[3] | 1 | (* *********************************************************************) |
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| 2 | (* *) |
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| 3 | (* The Compcert verified compiler *) |
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| 4 | (* *) |
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| 5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
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| 6 | (* *) |
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| 7 | (* Copyright Institut National de Recherche en Informatique et en *) |
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| 8 | (* Automatique. All rights reserved. This file is distributed *) |
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| 9 | (* under the terms of the GNU General Public License as published by *) |
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| 10 | (* the Free Software Foundation, either version 2 of the License, or *) |
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| 11 | (* (at your option) any later version. This file is also distributed *) |
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| 12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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| 13 | (* *) |
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| 14 | (* *********************************************************************) |
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| 15 | |
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| 16 | (* Formalizations of machine integers modulo $2^N$ #2<sup>N</sup>#. *) |
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| 17 | |
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| 18 | include "arithmetics/nat.ma". |
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[744] | 19 | include "utilities/extranat.ma". |
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[3] | 20 | |
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[700] | 21 | include "ASM/BitVector.ma". |
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| 22 | include "ASM/Arithmetic.ma". |
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[3] | 23 | |
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| 24 | (* * * Comparisons *) |
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| 25 | |
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[487] | 26 | inductive comparison : Type[0] ≝ |
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[3] | 27 | | Ceq : comparison (**r same *) |
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| 28 | | Cne : comparison (**r different *) |
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| 29 | | Clt : comparison (**r less than *) |
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| 30 | | Cle : comparison (**r less than or equal *) |
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| 31 | | Cgt : comparison (**r greater than *) |
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| 32 | | Cge : comparison. (**r greater than or equal *) |
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| 33 | |
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[487] | 34 | definition negate_comparison : comparison → comparison ≝ λc. |
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[3] | 35 | match c with |
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| 36 | [ Ceq ⇒ Cne |
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| 37 | | Cne ⇒ Ceq |
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| 38 | | Clt ⇒ Cge |
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| 39 | | Cle ⇒ Cgt |
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| 40 | | Cgt ⇒ Cle |
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| 41 | | Cge ⇒ Clt |
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| 42 | ]. |
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| 43 | |
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[487] | 44 | definition swap_comparison : comparison → comparison ≝ λc. |
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[3] | 45 | match c with |
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| 46 | [ Ceq ⇒ Ceq |
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| 47 | | Cne ⇒ Cne |
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| 48 | | Clt ⇒ Cgt |
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| 49 | | Cle ⇒ Cge |
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| 50 | | Cgt ⇒ Clt |
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| 51 | | Cge ⇒ Cle |
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| 52 | ]. |
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| 53 | (* |
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| 54 | (** * Parameterization by the word size, in bits. *) |
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| 55 | |
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| 56 | Module Type WORDSIZE. |
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| 57 | Variable wordsize: nat. |
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| 58 | Axiom wordsize_not_zero: wordsize <> 0%nat. |
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| 59 | End WORDSIZE. |
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| 60 | |
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| 61 | Module Make(WS: WORDSIZE). |
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| 62 | |
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| 63 | *) |
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| 64 | |
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[487] | 65 | (*axiom two_power_nat : nat → Z.*) |
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[3] | 66 | |
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[487] | 67 | definition wordsize : nat ≝ 32. |
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[744] | 68 | (* |
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[487] | 69 | definition modulus : Z ≝ Z_two_power_nat wordsize. |
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| 70 | definition half_modulus : Z ≝ modulus / 2. |
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| 71 | definition max_unsigned : Z ≝ modulus - 1. |
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| 72 | definition max_signed : Z ≝ half_modulus - 1. |
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| 73 | definition min_signed : Z ≝ - half_modulus. |
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[3] | 74 | |
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[487] | 75 | lemma wordsize_pos: |
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[3] | 76 | Z_of_nat wordsize > 0. |
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[487] | 77 | normalize; //; qed. |
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[3] | 78 | |
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[487] | 79 | lemma modulus_power: |
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[3] | 80 | modulus = two_p (Z_of_nat wordsize). |
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[487] | 81 | //; qed. |
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[3] | 82 | |
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[487] | 83 | lemma modulus_pos: |
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[3] | 84 | modulus > 0. |
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[487] | 85 | //; qed. |
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[744] | 86 | *) |
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[3] | 87 | (* * Representation of machine integers *) |
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| 88 | |
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| 89 | (* A machine integer (type [int]) is represented as a Coq arbitrary-precision |
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| 90 | integer (type [Z]) plus a proof that it is in the range 0 (included) to |
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| 91 | [modulus] (excluded. *) |
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[535] | 92 | |
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| 93 | definition int : Type[0] ≝ BitVector wordsize. |
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[744] | 94 | (* |
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[535] | 95 | definition intval: int → Z ≝ Z_of_unsigned_bitvector ?. |
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[487] | 96 | definition intrange: ∀i:int. 0 ≤ (intval i) ∧ (intval i) < modulus. |
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[535] | 97 | #i % whd in ⊢ (?%%) |
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[547] | 98 | [ @bv_Z_unsigned_min |
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| 99 | | @bv_Z_unsigned_max |
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[535] | 100 | ] qed. |
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[3] | 101 | |
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| 102 | (* The [unsigned] and [signed] functions return the Coq integer corresponding |
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| 103 | to the given machine integer, interpreted as unsigned or signed |
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| 104 | respectively. *) |
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| 105 | |
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[487] | 106 | definition unsigned : int → Z ≝ intval. |
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[3] | 107 | |
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[487] | 108 | definition signed : int → Z ≝ λn. |
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[3] | 109 | if Zltb (unsigned n) half_modulus |
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| 110 | then unsigned n |
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| 111 | else unsigned n - modulus. |
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[744] | 112 | *) |
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[3] | 113 | (* Conversely, [repr] takes a Coq integer and returns the corresponding |
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| 114 | machine integer. The argument is treated modulo [modulus]. *) |
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[744] | 115 | (* |
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[535] | 116 | definition repr : Z → int ≝ λz. bitvector_of_Z wordsize z. |
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[744] | 117 | *) |
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| 118 | definition repr : nat → int ≝ λn. bitvector_of_nat wordsize n. |
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[535] | 119 | |
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[487] | 120 | definition zero := repr 0. |
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| 121 | definition one := repr 1. |
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[744] | 122 | definition mone := subtraction ? zero one. |
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| 123 | definition iwordsize := repr wordsize. |
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[3] | 124 | |
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[535] | 125 | lemma eq_dec: ∀x,y: int. (x = y) + (x ≠ y). |
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[1516] | 126 | #x #y lapply (refl ? (eq_bv ? x y)) cases (eq_bv ? x y) in ⊢ (???% → ?); #E |
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[535] | 127 | [ %1 lapply E @(eq_bv_elim … x y) [ // | #_ #X destruct ] |
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| 128 | | %2 lapply E @(eq_bv_elim … x y) [ #_ #X destruct | /2/ ] |
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| 129 | ] qed. |
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[3] | 130 | |
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| 131 | (* * Arithmetic and logical operations over machine integers *) |
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| 132 | |
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[535] | 133 | definition eq : int → int → bool ≝ eq_bv wordsize. |
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[582] | 134 | definition lt : int → int → bool ≝ lt_s wordsize. |
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| 135 | definition ltu : int → int → bool ≝ lt_u wordsize. |
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[3] | 136 | |
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[648] | 137 | definition neg : int → int ≝ two_complement_negation wordsize. |
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[744] | 138 | definition mul ≝ λx,y. \snd (split ? wordsize wordsize (multiplication wordsize x y)). |
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[3] | 139 | |
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[744] | 140 | definition zero_ext_n : ∀w,n:nat. BitVector w → BitVector w ≝ |
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| 141 | λw,n. |
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| 142 | match nat_compare n w return λx,y.λ_. BitVector y → BitVector y with |
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| 143 | [ nat_lt n' w' ⇒ λi. |
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| 144 | let 〈h,l〉 ≝ split ? (S w') n' (switch_bv_plus ??? i) in |
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| 145 | switch_bv_plus ??? (pad ?? l) |
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| 146 | | _ ⇒ λi.i |
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| 147 | ]. |
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[14] | 148 | |
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[744] | 149 | definition zero_ext : nat → int → int ≝ zero_ext_n wordsize. |
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[3] | 150 | |
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[744] | 151 | definition sign_ext_n : ∀w,n:nat. BitVector w → BitVector w ≝ |
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| 152 | λw,n. |
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| 153 | match nat_compare n w return λx,y.λ_. BitVector y → BitVector y with |
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| 154 | [ nat_lt n' w' ⇒ λi. |
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| 155 | let 〈h,l〉 ≝ split ? (S w') n' (switch_bv_plus ??? i) in |
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| 156 | switch_bv_plus ??? (pad_vector ? (match l with [ VEmpty ⇒ false | VCons _ h _ ⇒ h ]) ?? l) |
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| 157 | | _ ⇒ λi.i |
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| 158 | ]. |
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[3] | 159 | |
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[744] | 160 | definition sign_ext : nat → int → int ≝ sign_ext_n wordsize. |
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[3] | 161 | |
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| 162 | (* * Bitwise logical ``and'', ``or'' and ``xor'' operations. *) |
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| 163 | |
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| 164 | |
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[582] | 165 | definition i_and : int → int → int ≝ conjunction_bv wordsize. |
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| 166 | definition or : int → int → int ≝ inclusive_disjunction_bv wordsize. |
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| 167 | definition xor : int → int → int ≝ exclusive_disjunction_bv wordsize. |
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[3] | 168 | |
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[582] | 169 | definition not : int → int ≝ negation_bv wordsize. |
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[3] | 170 | |
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| 171 | (* * Shifts and rotates. *) |
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| 172 | |
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[744] | 173 | definition shl : int → int → int ≝ λx,y. shift_left ?? (nat_of_bitvector … y) x false. |
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| 174 | definition shru : int → int → int ≝ λx,y. shift_right ?? (nat_of_bitvector … y) x false. |
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[3] | 175 | |
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| 176 | (* * Arithmetic right shift is defined as signed division by a power of two. |
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| 177 | Two such shifts are defined: [shr] rounds towards minus infinity |
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| 178 | (standard behaviour for arithmetic right shift) and |
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| 179 | [shrx] rounds towards zero. *) |
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[14] | 180 | |
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[744] | 181 | definition shr : int → int → int ≝ λx,y. shift_right ?? (nat_of_bitvector … y) x (head' … x). |
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[487] | 182 | definition shrx : int → int → int ≝ λx,y. |
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[744] | 183 | match division_s ? x (shl one y) with [ None ⇒ zero | Some i ⇒ i ]. |
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[3] | 184 | |
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[487] | 185 | definition shr_carry ≝ λx,y: int. |
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[744] | 186 | subtraction ? (shrx x y) (shr x y). |
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[3] | 187 | |
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[744] | 188 | definition rol : int → int → int ≝ λx,y. rotate_left ?? (nat_of_bitvector ? y) x. |
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| 189 | definition ror : int → int → int ≝ λx,y. rotate_right ?? (nat_of_bitvector ? y) x. |
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[3] | 190 | |
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[487] | 191 | definition rolm ≝ λx,a,m: int. i_and (rol x a) m. |
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[3] | 192 | (* |
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| 193 | (** Decomposition of a number as a sum of powers of two. *) |
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| 194 | |
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| 195 | Fixpoint Z_one_bits (n: nat) (x: Z) (i: Z) {struct n}: list Z := |
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| 196 | match n with |
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| 197 | | O => nil |
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| 198 | | S m => |
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| 199 | let (b, y) := Z_bin_decomp x in |
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| 200 | if b then i :: Z_one_bits m y (i+1) else Z_one_bits m y (i+1) |
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| 201 | end. |
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| 202 | |
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| 203 | Definition one_bits (x: int) : list int := |
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| 204 | List.map repr (Z_one_bits wordsize (unsigned x) 0). |
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| 205 | |
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| 206 | (** Recognition of powers of two. *) |
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| 207 | |
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| 208 | Definition is_power2 (x: int) : option int := |
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| 209 | match Z_one_bits wordsize (unsigned x) 0 with |
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| 210 | | i :: nil => Some (repr i) |
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| 211 | | _ => None |
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| 212 | end. |
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| 213 | |
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| 214 | (** Recognition of integers that are acceptable as immediate operands |
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| 215 | to the [rlwim] PowerPC instruction. These integers are of the form |
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| 216 | [000011110000] or [111100001111], that is, a run of one bits |
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| 217 | surrounded by zero bits, or conversely. We recognize these integers by |
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| 218 | running the following automaton on the bits. The accepting states are |
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| 219 | 2, 3, 4, 5, and 6. |
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| 220 | << |
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| 221 | 0 1 0 |
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| 222 | / \ / \ / \ |
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| 223 | \ / \ / \ / |
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| 224 | -0--> [1] --1--> [2] --0--> [3] |
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| 225 | / |
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| 226 | [0] |
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| 227 | \ |
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| 228 | -1--> [4] --0--> [5] --1--> [6] |
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| 229 | / \ / \ / \ |
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| 230 | \ / \ / \ / |
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| 231 | 1 0 1 |
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| 232 | >> |
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| 233 | *) |
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| 234 | |
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| 235 | Inductive rlw_state: Type := |
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| 236 | | RLW_S0 : rlw_state |
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| 237 | | RLW_S1 : rlw_state |
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| 238 | | RLW_S2 : rlw_state |
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| 239 | | RLW_S3 : rlw_state |
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| 240 | | RLW_S4 : rlw_state |
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| 241 | | RLW_S5 : rlw_state |
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| 242 | | RLW_S6 : rlw_state |
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| 243 | | RLW_Sbad : rlw_state. |
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| 244 | |
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| 245 | Definition rlw_transition (s: rlw_state) (b: bool) : rlw_state := |
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| 246 | match s, b with |
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| 247 | | RLW_S0, false => RLW_S1 |
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| 248 | | RLW_S0, true => RLW_S4 |
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| 249 | | RLW_S1, false => RLW_S1 |
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| 250 | | RLW_S1, true => RLW_S2 |
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| 251 | | RLW_S2, false => RLW_S3 |
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| 252 | | RLW_S2, true => RLW_S2 |
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| 253 | | RLW_S3, false => RLW_S3 |
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| 254 | | RLW_S3, true => RLW_Sbad |
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| 255 | | RLW_S4, false => RLW_S5 |
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| 256 | | RLW_S4, true => RLW_S4 |
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| 257 | | RLW_S5, false => RLW_S5 |
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| 258 | | RLW_S5, true => RLW_S6 |
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| 259 | | RLW_S6, false => RLW_Sbad |
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| 260 | | RLW_S6, true => RLW_S6 |
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| 261 | | RLW_Sbad, _ => RLW_Sbad |
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| 262 | end. |
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| 263 | |
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| 264 | Definition rlw_accepting (s: rlw_state) : bool := |
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| 265 | match s with |
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| 266 | | RLW_S0 => false |
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| 267 | | RLW_S1 => false |
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| 268 | | RLW_S2 => true |
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| 269 | | RLW_S3 => true |
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| 270 | | RLW_S4 => true |
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| 271 | | RLW_S5 => true |
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| 272 | | RLW_S6 => true |
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| 273 | | RLW_Sbad => false |
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| 274 | end. |
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| 275 | |
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| 276 | Fixpoint is_rlw_mask_rec (n: nat) (s: rlw_state) (x: Z) {struct n} : bool := |
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| 277 | match n with |
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| 278 | | O => |
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| 279 | rlw_accepting s |
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| 280 | | S m => |
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| 281 | let (b, y) := Z_bin_decomp x in |
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| 282 | is_rlw_mask_rec m (rlw_transition s b) y |
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| 283 | end. |
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| 284 | |
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| 285 | Definition is_rlw_mask (x: int) : bool := |
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| 286 | is_rlw_mask_rec wordsize RLW_S0 (unsigned x). |
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| 287 | *) |
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| 288 | (* * Comparisons. *) |
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| 289 | |
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[487] | 290 | definition cmp : comparison → int → int → bool ≝ λc,x,y. |
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[3] | 291 | match c with |
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| 292 | [ Ceq ⇒ eq x y |
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| 293 | | Cne ⇒ notb (eq x y) |
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| 294 | | Clt ⇒ lt x y |
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| 295 | | Cle ⇒ notb (lt y x) |
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| 296 | | Cgt ⇒ lt y x |
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| 297 | | Cge ⇒ notb (lt x y) |
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| 298 | ]. |
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| 299 | |
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[487] | 300 | definition cmpu : comparison → int → int → bool ≝ λc,x,y. |
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[3] | 301 | match c with |
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| 302 | [ Ceq ⇒ eq x y |
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| 303 | | Cne ⇒ notb (eq x y) |
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| 304 | | Clt ⇒ ltu x y |
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| 305 | | Cle ⇒ notb (ltu y x) |
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| 306 | | Cgt ⇒ ltu y x |
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| 307 | | Cge ⇒ notb (ltu x y) |
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| 308 | ]. |
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| 309 | |
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[487] | 310 | definition is_false : int → Prop ≝ λx. x = zero. |
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| 311 | definition is_true : int → Prop ≝ λx. x ≠ zero. |
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| 312 | definition notbool : int → int ≝ λx. if eq x zero then one else zero. |
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[3] | 313 | (* |
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| 314 | (** * Properties of integers and integer arithmetic *) |
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| 315 | |
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| 316 | (** ** Properties of [modulus], [max_unsigned], etc. *) |
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| 317 | |
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| 318 | Remark half_modulus_power: |
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| 319 | half_modulus = two_p (Z_of_nat wordsize - 1). |
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| 320 | Proof. |
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| 321 | unfold half_modulus. rewrite modulus_power. |
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| 322 | set (ws1 := Z_of_nat wordsize - 1). |
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| 323 | replace (Z_of_nat wordsize) with (Zsucc ws1). |
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| 324 | rewrite two_p_S. rewrite Zmult_comm. apply Z_div_mult. omega. |
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| 325 | unfold ws1. generalize wordsize_pos; omega. |
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| 326 | unfold ws1. omega. |
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| 327 | Qed. |
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| 328 | |
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| 329 | Remark half_modulus_modulus: modulus = 2 * half_modulus. |
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| 330 | Proof. |
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| 331 | rewrite half_modulus_power. rewrite modulus_power. |
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| 332 | rewrite <- two_p_S. decEq. omega. |
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| 333 | generalize wordsize_pos; omega. |
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| 334 | Qed. |
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| 335 | |
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| 336 | (** Relative positions, from greatest to smallest: |
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| 337 | << |
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| 338 | max_unsigned |
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| 339 | max_signed |
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| 340 | 2*wordsize-1 |
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| 341 | wordsize |
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| 342 | 0 |
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| 343 | min_signed |
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| 344 | >> |
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| 345 | *) |
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| 346 | |
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| 347 | Remark half_modulus_pos: half_modulus > 0. |
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| 348 | Proof. |
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| 349 | rewrite half_modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; omega. |
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| 350 | Qed. |
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| 351 | |
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| 352 | Remark min_signed_neg: min_signed < 0. |
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| 353 | Proof. |
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| 354 | unfold min_signed. generalize half_modulus_pos. omega. |
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| 355 | Qed. |
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| 356 | |
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| 357 | Remark max_signed_pos: max_signed >= 0. |
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| 358 | Proof. |
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| 359 | unfold max_signed. generalize half_modulus_pos. omega. |
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| 360 | Qed. |
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| 361 | |
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| 362 | Remark wordsize_max_unsigned: Z_of_nat wordsize <= max_unsigned. |
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| 363 | Proof. |
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| 364 | assert (Z_of_nat wordsize < modulus). |
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| 365 | rewrite modulus_power. apply two_p_strict. |
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| 366 | generalize wordsize_pos. omega. |
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| 367 | unfold max_unsigned. omega. |
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| 368 | Qed. |
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| 369 | |
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| 370 | Remark two_wordsize_max_unsigned: 2 * Z_of_nat wordsize - 1 <= max_unsigned. |
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| 371 | Proof. |
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| 372 | assert (2 * Z_of_nat wordsize - 1 < modulus). |
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| 373 | rewrite modulus_power. apply two_p_strict_2. generalize wordsize_pos; omega. |
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| 374 | unfold max_unsigned; omega. |
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| 375 | Qed. |
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| 376 | |
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| 377 | Remark max_signed_unsigned: max_signed < max_unsigned. |
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| 378 | Proof. |
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| 379 | unfold max_signed, max_unsigned. rewrite half_modulus_modulus. |
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| 380 | generalize half_modulus_pos. omega. |
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| 381 | Qed. |
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| 382 | |
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| 383 | (** ** Properties of zero, one, minus one *) |
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| 384 | |
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| 385 | Theorem unsigned_zero: unsigned zero = 0. |
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| 386 | Proof. |
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| 387 | simpl. apply Zmod_0_l. |
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| 388 | Qed. |
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| 389 | |
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| 390 | Theorem unsigned_one: unsigned one = 1. |
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| 391 | Proof. |
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| 392 | simpl. apply Zmod_small. split. omega. |
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| 393 | unfold modulus. replace wordsize with (S(pred wordsize)). |
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| 394 | rewrite two_power_nat_S. generalize (two_power_nat_pos (pred wordsize)). |
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| 395 | omega. |
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| 396 | generalize wordsize_pos. omega. |
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| 397 | Qed. |
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| 398 | |
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| 399 | Theorem unsigned_mone: unsigned mone = modulus - 1. |
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| 400 | Proof. |
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| 401 | simpl unsigned. |
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| 402 | replace (-1) with ((modulus - 1) + (-1) * modulus). |
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| 403 | rewrite Z_mod_plus_full. apply Zmod_small. |
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| 404 | generalize modulus_pos. omega. omega. |
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| 405 | Qed. |
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| 406 | |
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| 407 | Theorem signed_zero: signed zero = 0. |
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| 408 | Proof. |
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| 409 | unfold signed. rewrite unsigned_zero. apply zlt_true. generalize half_modulus_pos; omega. |
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| 410 | Qed. |
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| 411 | |
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| 412 | Theorem signed_mone: signed mone = -1. |
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| 413 | Proof. |
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| 414 | unfold signed. rewrite unsigned_mone. |
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| 415 | rewrite zlt_false. omega. |
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| 416 | rewrite half_modulus_modulus. generalize half_modulus_pos. omega. |
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| 417 | Qed. |
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[4] | 418 | *) |
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[582] | 419 | theorem one_not_zero: one ≠ zero. |
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[744] | 420 | % #H @(match eq_dec one zero return λx.match x with [ inl _ ⇒ True | inr _ ⇒ False ] with [ inl _ ⇒ I | inr p ⇒ ?]) normalize |
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| 421 | cases p #H' @(H' H) |
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[582] | 422 | qed. |
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| 423 | |
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[4] | 424 | (* |
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[3] | 425 | Theorem unsigned_repr_wordsize: |
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| 426 | unsigned iwordsize = Z_of_nat wordsize. |
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| 427 | Proof. |
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| 428 | simpl. apply Zmod_small. |
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| 429 | generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; omega. |
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| 430 | Qed. |
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[4] | 431 | *) |
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| 432 | (* * ** Properties of equality *) |
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[3] | 433 | |
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[487] | 434 | theorem eq_sym: |
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[4] | 435 | ∀x,y. eq x y = eq y x. |
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[889] | 436 | #x #y change with (eq_bv ??? = eq_bv ???) |
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[1512] | 437 | @eq_bv_elim @eq_bv_elim /2/ |
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[961] | 438 | qed. |
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[3] | 439 | |
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[487] | 440 | theorem eq_spec: ∀x,y: int. if eq x y then x = y else (x ≠ y). |
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[889] | 441 | #x #y change with (if eq_bv ? x y then ? else ?) @eq_bv_elim #H @H qed. |
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[3] | 442 | |
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[487] | 443 | theorem eq_true: ∀x. eq x x = true. |
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| 444 | #x lapply (eq_spec x x); elim (eq x x); //; |
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| 445 | #H normalize in H; @False_ind @(absurd ? (refl ??) H) |
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| 446 | qed. |
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[3] | 447 | |
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[487] | 448 | theorem eq_false: ∀x,y. x ≠ y → eq x y = false. |
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| 449 | #x #y lapply (eq_spec x y); elim (eq x y); //; |
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| 450 | #H #H' @False_ind @(absurd ? H H') |
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| 451 | qed. |
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[4] | 452 | (* |
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[3] | 453 | (** ** Modulo arithmetic *) |
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| 454 | |
---|
| 455 | (** We define and state properties of equality and arithmetic modulo a |
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| 456 | positive integer. *) |
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| 457 | |
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| 458 | Section EQ_MODULO. |
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| 459 | |
---|
| 460 | Variable modul: Z. |
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| 461 | Hypothesis modul_pos: modul > 0. |
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| 462 | |
---|
| 463 | Definition eqmod (x y: Z) : Prop := exists k, x = k * modul + y. |
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| 464 | |
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| 465 | Lemma eqmod_refl: forall x, eqmod x x. |
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| 466 | Proof. |
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| 467 | intros; red. exists 0. omega. |
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| 468 | Qed. |
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| 469 | |
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| 470 | Lemma eqmod_refl2: forall x y, x = y -> eqmod x y. |
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| 471 | Proof. |
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| 472 | intros. subst y. apply eqmod_refl. |
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| 473 | Qed. |
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| 474 | |
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| 475 | Lemma eqmod_sym: forall x y, eqmod x y -> eqmod y x. |
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| 476 | Proof. |
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| 477 | intros x y [k EQ]; red. exists (-k). subst x. ring. |
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| 478 | Qed. |
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| 479 | |
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| 480 | Lemma eqmod_trans: forall x y z, eqmod x y -> eqmod y z -> eqmod x z. |
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| 481 | Proof. |
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| 482 | intros x y z [k1 EQ1] [k2 EQ2]; red. |
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| 483 | exists (k1 + k2). subst x; subst y. ring. |
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| 484 | Qed. |
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| 485 | |
---|
| 486 | Lemma eqmod_small_eq: |
---|
| 487 | forall x y, eqmod x y -> 0 <= x < modul -> 0 <= y < modul -> x = y. |
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| 488 | Proof. |
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| 489 | intros x y [k EQ] I1 I2. |
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| 490 | generalize (Zdiv_unique _ _ _ _ EQ I2). intro. |
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| 491 | rewrite (Zdiv_small x modul I1) in H. subst k. omega. |
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| 492 | Qed. |
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| 493 | |
---|
| 494 | Lemma eqmod_mod_eq: |
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| 495 | forall x y, eqmod x y -> x mod modul = y mod modul. |
---|
| 496 | Proof. |
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| 497 | intros x y [k EQ]. subst x. |
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| 498 | rewrite Zplus_comm. apply Z_mod_plus. auto. |
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| 499 | Qed. |
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| 500 | |
---|
| 501 | Lemma eqmod_mod: |
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| 502 | forall x, eqmod x (x mod modul). |
---|
| 503 | Proof. |
---|
| 504 | intros; red. exists (x / modul). |
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| 505 | rewrite Zmult_comm. apply Z_div_mod_eq. auto. |
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| 506 | Qed. |
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| 507 | |
---|
| 508 | Lemma eqmod_add: |
---|
| 509 | forall a b c d, eqmod a b -> eqmod c d -> eqmod (a + c) (b + d). |
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| 510 | Proof. |
---|
| 511 | intros a b c d [k1 EQ1] [k2 EQ2]; red. |
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| 512 | subst a; subst c. exists (k1 + k2). ring. |
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| 513 | Qed. |
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| 514 | |
---|
| 515 | Lemma eqmod_neg: |
---|
| 516 | forall x y, eqmod x y -> eqmod (-x) (-y). |
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| 517 | Proof. |
---|
| 518 | intros x y [k EQ]; red. exists (-k). rewrite EQ. ring. |
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| 519 | Qed. |
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| 520 | |
---|
| 521 | Lemma eqmod_sub: |
---|
| 522 | forall a b c d, eqmod a b -> eqmod c d -> eqmod (a - c) (b - d). |
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| 523 | Proof. |
---|
| 524 | intros a b c d [k1 EQ1] [k2 EQ2]; red. |
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| 525 | subst a; subst c. exists (k1 - k2). ring. |
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| 526 | Qed. |
---|
| 527 | |
---|
| 528 | Lemma eqmod_mult: |
---|
| 529 | forall a b c d, eqmod a c -> eqmod b d -> eqmod (a * b) (c * d). |
---|
| 530 | Proof. |
---|
| 531 | intros a b c d [k1 EQ1] [k2 EQ2]; red. |
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| 532 | subst a; subst b. |
---|
| 533 | exists (k1 * k2 * modul + c * k2 + k1 * d). |
---|
| 534 | ring. |
---|
| 535 | Qed. |
---|
| 536 | |
---|
| 537 | End EQ_MODULO. |
---|
| 538 | |
---|
| 539 | Lemma eqmod_divides: |
---|
| 540 | forall n m x y, eqmod n x y -> Zdivide m n -> eqmod m x y. |
---|
| 541 | Proof. |
---|
| 542 | intros. destruct H as [k1 EQ1]. destruct H0 as [k2 EQ2]. |
---|
| 543 | exists (k1*k2). rewrite <- Zmult_assoc. rewrite <- EQ2. auto. |
---|
| 544 | Qed. |
---|
| 545 | |
---|
| 546 | (** We then specialize these definitions to equality modulo |
---|
| 547 | $2^{wordsize}$ #2<sup>wordsize</sup>#. *) |
---|
| 548 | |
---|
| 549 | Hint Resolve modulus_pos: ints. |
---|
| 550 | |
---|
| 551 | Definition eqm := eqmod modulus. |
---|
| 552 | |
---|
| 553 | Lemma eqm_refl: forall x, eqm x x. |
---|
| 554 | Proof (eqmod_refl modulus). |
---|
| 555 | Hint Resolve eqm_refl: ints. |
---|
| 556 | |
---|
| 557 | Lemma eqm_refl2: |
---|
| 558 | forall x y, x = y -> eqm x y. |
---|
| 559 | Proof (eqmod_refl2 modulus). |
---|
| 560 | Hint Resolve eqm_refl2: ints. |
---|
| 561 | |
---|
| 562 | Lemma eqm_sym: forall x y, eqm x y -> eqm y x. |
---|
| 563 | Proof (eqmod_sym modulus). |
---|
| 564 | Hint Resolve eqm_sym: ints. |
---|
| 565 | |
---|
| 566 | Lemma eqm_trans: forall x y z, eqm x y -> eqm y z -> eqm x z. |
---|
| 567 | Proof (eqmod_trans modulus). |
---|
| 568 | Hint Resolve eqm_trans: ints. |
---|
| 569 | |
---|
| 570 | Lemma eqm_samerepr: forall x y, eqm x y -> repr x = repr y. |
---|
| 571 | Proof. |
---|
| 572 | intros. unfold repr. apply mkint_eq. |
---|
| 573 | apply eqmod_mod_eq. auto with ints. exact H. |
---|
| 574 | Qed. |
---|
| 575 | |
---|
| 576 | Lemma eqm_small_eq: |
---|
| 577 | forall x y, eqm x y -> 0 <= x < modulus -> 0 <= y < modulus -> x = y. |
---|
| 578 | Proof (eqmod_small_eq modulus). |
---|
| 579 | Hint Resolve eqm_small_eq: ints. |
---|
| 580 | |
---|
| 581 | Lemma eqm_add: |
---|
| 582 | forall a b c d, eqm a b -> eqm c d -> eqm (a + c) (b + d). |
---|
| 583 | Proof (eqmod_add modulus). |
---|
| 584 | Hint Resolve eqm_add: ints. |
---|
| 585 | |
---|
| 586 | Lemma eqm_neg: |
---|
| 587 | forall x y, eqm x y -> eqm (-x) (-y). |
---|
| 588 | Proof (eqmod_neg modulus). |
---|
| 589 | Hint Resolve eqm_neg: ints. |
---|
| 590 | |
---|
| 591 | Lemma eqm_sub: |
---|
| 592 | forall a b c d, eqm a b -> eqm c d -> eqm (a - c) (b - d). |
---|
| 593 | Proof (eqmod_sub modulus). |
---|
| 594 | Hint Resolve eqm_sub: ints. |
---|
| 595 | |
---|
| 596 | Lemma eqm_mult: |
---|
| 597 | forall a b c d, eqm a c -> eqm b d -> eqm (a * b) (c * d). |
---|
| 598 | Proof (eqmod_mult modulus). |
---|
| 599 | Hint Resolve eqm_mult: ints. |
---|
| 600 | |
---|
| 601 | (** ** Properties of the coercions between [Z] and [int] *) |
---|
| 602 | |
---|
| 603 | Lemma eqm_unsigned_repr: |
---|
| 604 | forall z, eqm z (unsigned (repr z)). |
---|
| 605 | Proof. |
---|
| 606 | unfold eqm, repr, unsigned; intros; simpl. |
---|
| 607 | apply eqmod_mod. auto with ints. |
---|
| 608 | Qed. |
---|
| 609 | Hint Resolve eqm_unsigned_repr: ints. |
---|
| 610 | |
---|
| 611 | Lemma eqm_unsigned_repr_l: |
---|
| 612 | forall a b, eqm a b -> eqm (unsigned (repr a)) b. |
---|
| 613 | Proof. |
---|
| 614 | intros. apply eqm_trans with a. |
---|
| 615 | apply eqm_sym. apply eqm_unsigned_repr. auto. |
---|
| 616 | Qed. |
---|
| 617 | Hint Resolve eqm_unsigned_repr_l: ints. |
---|
| 618 | |
---|
| 619 | Lemma eqm_unsigned_repr_r: |
---|
| 620 | forall a b, eqm a b -> eqm a (unsigned (repr b)). |
---|
| 621 | Proof. |
---|
| 622 | intros. apply eqm_trans with b. auto. |
---|
| 623 | apply eqm_unsigned_repr. |
---|
| 624 | Qed. |
---|
| 625 | Hint Resolve eqm_unsigned_repr_r: ints. |
---|
| 626 | |
---|
| 627 | Lemma eqm_signed_unsigned: |
---|
| 628 | forall x, eqm (signed x) (unsigned x). |
---|
| 629 | Proof. |
---|
| 630 | intro; red; unfold signed. set (y := unsigned x). |
---|
| 631 | case (zlt y half_modulus); intro. |
---|
| 632 | apply eqmod_refl. red; exists (-1); ring. |
---|
| 633 | Qed. |
---|
[181] | 634 | *) |
---|
[744] | 635 | (* |
---|
[487] | 636 | theorem unsigned_range: ∀i. 0 ≤ unsigned i ∧ unsigned i < modulus. |
---|
[535] | 637 | #i @intrange |
---|
[487] | 638 | qed. |
---|
[3] | 639 | |
---|
[487] | 640 | theorem unsigned_range_2: |
---|
[181] | 641 | ∀i. 0 ≤ unsigned i ∧ unsigned i ≤ max_unsigned. |
---|
[487] | 642 | #i >(?:max_unsigned = modulus - 1) //; (* unfold *) |
---|
| 643 | lapply (unsigned_range i); *; #Hz #Hm % |
---|
| 644 | [ //; |
---|
| 645 | | <(Zpred_Zsucc (unsigned i)) |
---|
| 646 | <(Zpred_Zplus_neg_O modulus) |
---|
| 647 | @monotonic_Zle_Zpred |
---|
[181] | 648 | /2/; |
---|
[487] | 649 | ] qed. |
---|
[181] | 650 | |
---|
[487] | 651 | axiom signed_range: |
---|
[3] | 652 | ∀i. min_signed ≤ signed i ∧ signed i ≤ max_signed. |
---|
| 653 | (* |
---|
[487] | 654 | #i whd in ⊢ (?(??%)(?%?)); |
---|
| 655 | lapply (unsigned_range i); *; letin n ≝ (unsigned i); #H1 #H2 |
---|
| 656 | @(Zltb_elim_Type0) #H3 |
---|
| 657 | [ % [ @(transitive_Zle ? OZ) //; |
---|
| 658 | | <(Zpred_Zsucc n) |
---|
| 659 | <(Zpred_Zplus_neg_O half_modulus) |
---|
| 660 | @monotonic_Zle_Zpred /2/; |
---|
| 661 | ] |
---|
| 662 | | % [ >half_modulus_modulus |
---|
[181] | 663 | |
---|
[3] | 664 | Theorem signed_range: |
---|
| 665 | forall i, min_signed <= signed i <= max_signed. |
---|
| 666 | Proof. |
---|
| 667 | intros. unfold signed. |
---|
| 668 | generalize (unsigned_range i). set (n := unsigned i). intros. |
---|
| 669 | case (zlt n half_modulus); intro. |
---|
| 670 | unfold max_signed. generalize min_signed_neg. omega. |
---|
| 671 | unfold min_signed, max_signed. |
---|
| 672 | rewrite half_modulus_modulus in *. omega. |
---|
| 673 | Qed. |
---|
| 674 | |
---|
| 675 | Theorem repr_unsigned: |
---|
| 676 | forall i, repr (unsigned i) = i. |
---|
| 677 | Proof. |
---|
| 678 | destruct i; simpl. unfold repr. apply mkint_eq. |
---|
| 679 | apply Zmod_small. auto. |
---|
| 680 | Qed. |
---|
| 681 | Hint Resolve repr_unsigned: ints. |
---|
| 682 | |
---|
| 683 | Lemma repr_signed: |
---|
| 684 | forall i, repr (signed i) = i. |
---|
| 685 | Proof. |
---|
| 686 | intros. transitivity (repr (unsigned i)). |
---|
| 687 | apply eqm_samerepr. apply eqm_signed_unsigned. auto with ints. |
---|
| 688 | Qed. |
---|
| 689 | Hint Resolve repr_signed: ints. |
---|
| 690 | |
---|
| 691 | Theorem unsigned_repr: |
---|
| 692 | forall z, 0 <= z <= max_unsigned -> unsigned (repr z) = z. |
---|
| 693 | Proof. |
---|
| 694 | intros. unfold repr, unsigned; simpl. |
---|
| 695 | apply Zmod_small. unfold max_unsigned in H. omega. |
---|
| 696 | Qed. |
---|
| 697 | Hint Resolve unsigned_repr: ints. |
---|
| 698 | *) |
---|
[487] | 699 | axiom signed_repr: |
---|
[3] | 700 | ∀z. min_signed ≤ z ∧ z ≤ max_signed → signed (repr z) = z. |
---|
| 701 | (* |
---|
| 702 | Theorem signed_repr: |
---|
| 703 | forall z, min_signed <= z <= max_signed -> signed (repr z) = z. |
---|
| 704 | Proof. |
---|
| 705 | intros. unfold signed. case (zle 0 z); intro. |
---|
| 706 | replace (unsigned (repr z)) with z. |
---|
| 707 | rewrite zlt_true. auto. unfold max_signed in H. omega. |
---|
| 708 | symmetry. apply unsigned_repr. generalize max_signed_unsigned. omega. |
---|
| 709 | pose (z' := z + modulus). |
---|
| 710 | replace (repr z) with (repr z'). |
---|
| 711 | replace (unsigned (repr z')) with z'. |
---|
| 712 | rewrite zlt_false. unfold z'. omega. |
---|
| 713 | unfold z'. unfold min_signed in H. |
---|
| 714 | rewrite half_modulus_modulus. omega. |
---|
| 715 | symmetry. apply unsigned_repr. |
---|
| 716 | unfold z', max_unsigned. unfold min_signed, max_signed in H. |
---|
| 717 | rewrite half_modulus_modulus. omega. |
---|
| 718 | apply eqm_samerepr. unfold z'; red. exists 1. omega. |
---|
| 719 | Qed. |
---|
| 720 | |
---|
| 721 | Theorem signed_eq_unsigned: |
---|
| 722 | forall x, unsigned x <= max_signed -> signed x = unsigned x. |
---|
| 723 | Proof. |
---|
| 724 | intros. unfold signed. destruct (zlt (unsigned x) half_modulus). |
---|
| 725 | auto. unfold max_signed in H. omegaContradiction. |
---|
| 726 | Qed. |
---|
| 727 | |
---|
| 728 | (** ** Properties of addition *) |
---|
| 729 | |
---|
| 730 | *) |
---|
[487] | 731 | axiom add_unsigned: ∀x,y. add x y = repr (unsigned x + unsigned y). |
---|
| 732 | axiom add_signed: ∀x,y. add x y = repr (signed x + signed y). |
---|
| 733 | axiom add_zero: ∀x. add x zero = x. |
---|
[3] | 734 | |
---|
| 735 | (* |
---|
| 736 | Theorem add_unsigned: forall x y, add x y = repr (unsigned x + unsigned y). |
---|
| 737 | Proof. intros; reflexivity. |
---|
| 738 | Qed. |
---|
| 739 | |
---|
| 740 | Theorem add_signed: forall x y, add x y = repr (signed x + signed y). |
---|
| 741 | Proof. |
---|
| 742 | intros. rewrite add_unsigned. apply eqm_samerepr. |
---|
| 743 | apply eqm_add; apply eqm_sym; apply eqm_signed_unsigned. |
---|
| 744 | Qed. |
---|
| 745 | |
---|
| 746 | Theorem add_commut: forall x y, add x y = add y x. |
---|
| 747 | Proof. intros; unfold add. decEq. omega. Qed. |
---|
| 748 | |
---|
| 749 | Theorem add_zero: forall x, add x zero = x. |
---|
| 750 | Proof. |
---|
| 751 | intros; unfold add, zero. change (unsigned (repr 0)) with 0. |
---|
| 752 | rewrite Zplus_0_r. apply repr_unsigned. |
---|
| 753 | Qed. |
---|
| 754 | |
---|
| 755 | Theorem add_assoc: forall x y z, add (add x y) z = add x (add y z). |
---|
| 756 | Proof. |
---|
| 757 | intros; unfold add. |
---|
| 758 | set (x' := unsigned x). |
---|
| 759 | set (y' := unsigned y). |
---|
| 760 | set (z' := unsigned z). |
---|
| 761 | apply eqm_samerepr. |
---|
| 762 | apply eqm_trans with ((x' + y') + z'). |
---|
| 763 | auto with ints. |
---|
| 764 | rewrite <- Zplus_assoc. auto with ints. |
---|
| 765 | Qed. |
---|
| 766 | |
---|
| 767 | Theorem add_permut: forall x y z, add x (add y z) = add y (add x z). |
---|
| 768 | Proof. |
---|
| 769 | intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut. |
---|
| 770 | Qed. |
---|
| 771 | |
---|
| 772 | Theorem add_neg_zero: forall x, add x (neg x) = zero. |
---|
| 773 | Proof. |
---|
| 774 | intros; unfold add, neg, zero. apply eqm_samerepr. |
---|
| 775 | replace 0 with (unsigned x + (- (unsigned x))). |
---|
| 776 | auto with ints. omega. |
---|
| 777 | Qed. |
---|
| 778 | |
---|
| 779 | (** ** Properties of negation *) |
---|
| 780 | |
---|
| 781 | Theorem neg_repr: forall z, neg (repr z) = repr (-z). |
---|
| 782 | Proof. |
---|
| 783 | intros; unfold neg. apply eqm_samerepr. auto with ints. |
---|
| 784 | Qed. |
---|
| 785 | |
---|
| 786 | Theorem neg_zero: neg zero = zero. |
---|
| 787 | Proof. |
---|
| 788 | unfold neg, zero. compute. apply mkint_eq. auto. |
---|
| 789 | Qed. |
---|
| 790 | |
---|
| 791 | Theorem neg_involutive: forall x, neg (neg x) = x. |
---|
| 792 | Proof. |
---|
| 793 | intros; unfold neg. transitivity (repr (unsigned x)). |
---|
| 794 | apply eqm_samerepr. apply eqm_trans with (- (- (unsigned x))). |
---|
| 795 | apply eqm_neg. apply eqm_unsigned_repr_l. apply eqm_refl. |
---|
| 796 | apply eqm_refl2. omega. apply repr_unsigned. |
---|
| 797 | Qed. |
---|
| 798 | |
---|
| 799 | Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y). |
---|
| 800 | Proof. |
---|
| 801 | intros; unfold neg, add. apply eqm_samerepr. |
---|
| 802 | apply eqm_trans with (- (unsigned x + unsigned y)). |
---|
| 803 | auto with ints. |
---|
| 804 | replace (- (unsigned x + unsigned y)) |
---|
| 805 | with ((- unsigned x) + (- unsigned y)). |
---|
| 806 | auto with ints. omega. |
---|
| 807 | Qed. |
---|
| 808 | |
---|
| 809 | (** ** Properties of subtraction *) |
---|
| 810 | |
---|
| 811 | Theorem sub_zero_l: forall x, sub x zero = x. |
---|
| 812 | Proof. |
---|
| 813 | intros; unfold sub. change (unsigned zero) with 0. |
---|
| 814 | replace (unsigned x - 0) with (unsigned x). apply repr_unsigned. |
---|
| 815 | omega. |
---|
| 816 | Qed. |
---|
| 817 | |
---|
| 818 | Theorem sub_zero_r: forall x, sub zero x = neg x. |
---|
| 819 | Proof. |
---|
| 820 | intros; unfold sub, neg. change (unsigned zero) with 0. |
---|
| 821 | replace (0 - unsigned x) with (- unsigned x). auto. |
---|
| 822 | omega. |
---|
| 823 | Qed. |
---|
| 824 | |
---|
| 825 | Theorem sub_add_opp: forall x y, sub x y = add x (neg y). |
---|
| 826 | Proof. |
---|
| 827 | intros; unfold sub, add, neg. |
---|
| 828 | replace (unsigned x - unsigned y) |
---|
| 829 | with (unsigned x + (- unsigned y)). |
---|
| 830 | apply eqm_samerepr. auto with ints. omega. |
---|
| 831 | Qed. |
---|
| 832 | |
---|
| 833 | Theorem sub_idem: forall x, sub x x = zero. |
---|
| 834 | Proof. |
---|
| 835 | intros; unfold sub. replace (unsigned x - unsigned x) with 0. |
---|
| 836 | reflexivity. omega. |
---|
| 837 | Qed. |
---|
| 838 | |
---|
| 839 | Theorem sub_add_l: forall x y z, sub (add x y) z = add (sub x z) y. |
---|
| 840 | Proof. |
---|
| 841 | intros. repeat rewrite sub_add_opp. |
---|
| 842 | repeat rewrite add_assoc. decEq. apply add_commut. |
---|
| 843 | Qed. |
---|
| 844 | |
---|
| 845 | Theorem sub_add_r: forall x y z, sub x (add y z) = add (sub x z) (neg y). |
---|
| 846 | Proof. |
---|
| 847 | intros. repeat rewrite sub_add_opp. |
---|
| 848 | rewrite neg_add_distr. rewrite add_permut. apply add_commut. |
---|
| 849 | Qed. |
---|
| 850 | |
---|
| 851 | Theorem sub_shifted: |
---|
| 852 | forall x y z, |
---|
| 853 | sub (add x z) (add y z) = sub x y. |
---|
| 854 | Proof. |
---|
| 855 | intros. rewrite sub_add_opp. rewrite neg_add_distr. |
---|
| 856 | rewrite add_assoc. |
---|
| 857 | rewrite (add_commut (neg y) (neg z)). |
---|
| 858 | rewrite <- (add_assoc z). rewrite add_neg_zero. |
---|
| 859 | rewrite (add_commut zero). rewrite add_zero. |
---|
| 860 | symmetry. apply sub_add_opp. |
---|
| 861 | Qed. |
---|
| 862 | |
---|
| 863 | Theorem sub_signed: |
---|
| 864 | forall x y, sub x y = repr (signed x - signed y). |
---|
| 865 | Proof. |
---|
| 866 | intros. unfold sub. apply eqm_samerepr. |
---|
| 867 | apply eqm_sub; apply eqm_sym; apply eqm_signed_unsigned. |
---|
| 868 | Qed. |
---|
| 869 | |
---|
| 870 | (** ** Properties of multiplication *) |
---|
| 871 | |
---|
| 872 | Theorem mul_commut: forall x y, mul x y = mul y x. |
---|
| 873 | Proof. |
---|
| 874 | intros; unfold mul. decEq. ring. |
---|
| 875 | Qed. |
---|
| 876 | |
---|
| 877 | Theorem mul_zero: forall x, mul x zero = zero. |
---|
| 878 | Proof. |
---|
| 879 | intros; unfold mul. change (unsigned zero) with 0. |
---|
| 880 | unfold zero. decEq. ring. |
---|
| 881 | Qed. |
---|
| 882 | |
---|
| 883 | Theorem mul_one: forall x, mul x one = x. |
---|
| 884 | Proof. |
---|
| 885 | intros; unfold mul. rewrite unsigned_one. |
---|
| 886 | transitivity (repr (unsigned x)). decEq. ring. |
---|
| 887 | apply repr_unsigned. |
---|
| 888 | Qed. |
---|
| 889 | |
---|
| 890 | Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z). |
---|
| 891 | Proof. |
---|
| 892 | intros; unfold mul. |
---|
| 893 | set (x' := unsigned x). |
---|
| 894 | set (y' := unsigned y). |
---|
| 895 | set (z' := unsigned z). |
---|
| 896 | apply eqm_samerepr. apply eqm_trans with ((x' * y') * z'). |
---|
| 897 | auto with ints. |
---|
| 898 | rewrite <- Zmult_assoc. auto with ints. |
---|
| 899 | Qed. |
---|
| 900 | |
---|
| 901 | Theorem mul_add_distr_l: |
---|
| 902 | forall x y z, mul (add x y) z = add (mul x z) (mul y z). |
---|
| 903 | Proof. |
---|
| 904 | intros; unfold mul, add. |
---|
| 905 | apply eqm_samerepr. |
---|
| 906 | set (x' := unsigned x). |
---|
| 907 | set (y' := unsigned y). |
---|
| 908 | set (z' := unsigned z). |
---|
| 909 | apply eqm_trans with ((x' + y') * z'). |
---|
| 910 | auto with ints. |
---|
| 911 | replace ((x' + y') * z') with (x' * z' + y' * z'). |
---|
| 912 | auto with ints. |
---|
| 913 | ring. |
---|
| 914 | Qed. |
---|
| 915 | |
---|
| 916 | Theorem mul_add_distr_r: |
---|
| 917 | forall x y z, mul x (add y z) = add (mul x y) (mul x z). |
---|
| 918 | Proof. |
---|
| 919 | intros. rewrite mul_commut. rewrite mul_add_distr_l. |
---|
| 920 | decEq; apply mul_commut. |
---|
| 921 | Qed. |
---|
| 922 | |
---|
| 923 | Theorem neg_mul_distr_l: |
---|
| 924 | forall x y, neg(mul x y) = mul (neg x) y. |
---|
| 925 | Proof. |
---|
| 926 | intros. unfold mul, neg. |
---|
| 927 | set (x' := unsigned x). set (y' := unsigned y). |
---|
| 928 | apply eqm_samerepr. apply eqm_trans with (- (x' * y')). |
---|
| 929 | auto with ints. |
---|
| 930 | replace (- (x' * y')) with ((-x') * y') by ring. |
---|
| 931 | auto with ints. |
---|
| 932 | Qed. |
---|
| 933 | |
---|
| 934 | Theorem neg_mul_distr_r: |
---|
| 935 | forall x y, neg(mul x y) = mul x (neg y). |
---|
| 936 | Proof. |
---|
| 937 | intros. rewrite (mul_commut x y). rewrite (mul_commut x (neg y)). |
---|
| 938 | apply neg_mul_distr_l. |
---|
| 939 | Qed. |
---|
| 940 | |
---|
| 941 | Theorem mul_signed: |
---|
| 942 | forall x y, mul x y = repr (signed x * signed y). |
---|
| 943 | Proof. |
---|
| 944 | intros; unfold mul. apply eqm_samerepr. |
---|
| 945 | apply eqm_mult; apply eqm_sym; apply eqm_signed_unsigned. |
---|
| 946 | Qed. |
---|
| 947 | |
---|
| 948 | (** ** Properties of binary decompositions *) |
---|
| 949 | |
---|
| 950 | Lemma Z_shift_add_bin_decomp: |
---|
| 951 | forall x, |
---|
| 952 | Z_shift_add (fst (Z_bin_decomp x)) (snd (Z_bin_decomp x)) = x. |
---|
| 953 | Proof. |
---|
| 954 | destruct x; simpl. |
---|
| 955 | auto. |
---|
| 956 | destruct p; reflexivity. |
---|
| 957 | destruct p; try reflexivity. simpl. |
---|
| 958 | assert (forall z, 2 * (z + 1) - 1 = 2 * z + 1). intro; omega. |
---|
| 959 | generalize (H (Zpos p)); simpl. congruence. |
---|
| 960 | Qed. |
---|
| 961 | |
---|
| 962 | Lemma Z_shift_add_inj: |
---|
| 963 | forall b1 x1 b2 x2, |
---|
| 964 | Z_shift_add b1 x1 = Z_shift_add b2 x2 -> b1 = b2 /\ x1 = x2. |
---|
| 965 | Proof. |
---|
| 966 | intros until x2. |
---|
| 967 | unfold Z_shift_add. |
---|
| 968 | destruct b1; destruct b2; intros; |
---|
| 969 | ((split; [reflexivity|omega]) || omegaContradiction). |
---|
| 970 | Qed. |
---|
| 971 | |
---|
| 972 | Lemma Z_of_bits_exten: |
---|
| 973 | forall n f1 f2, |
---|
| 974 | (forall z, 0 <= z < Z_of_nat n -> f1 z = f2 z) -> |
---|
| 975 | Z_of_bits n f1 = Z_of_bits n f2. |
---|
| 976 | Proof. |
---|
| 977 | induction n; intros. |
---|
| 978 | reflexivity. |
---|
| 979 | simpl. rewrite inj_S in H. decEq. apply H. omega. |
---|
| 980 | apply IHn. intros; apply H. omega. |
---|
| 981 | Qed. |
---|
| 982 | |
---|
| 983 | Opaque Zmult. |
---|
| 984 | |
---|
| 985 | Lemma Z_of_bits_of_Z: |
---|
| 986 | forall x, eqm (Z_of_bits wordsize (bits_of_Z wordsize x)) x. |
---|
| 987 | Proof. |
---|
| 988 | assert (forall n x, exists k, |
---|
| 989 | Z_of_bits n (bits_of_Z n x) = k * two_power_nat n + x). |
---|
| 990 | induction n; intros. |
---|
| 991 | rewrite two_power_nat_O. simpl. exists (-x). omega. |
---|
| 992 | rewrite two_power_nat_S. simpl. |
---|
| 993 | caseEq (Z_bin_decomp x). intros b y ZBD. simpl. |
---|
| 994 | replace (Z_of_bits n (fun i => if zeq (i + 1) 0 then b else bits_of_Z n y (i + 1 - 1))) |
---|
| 995 | with (Z_of_bits n (bits_of_Z n y)). |
---|
| 996 | elim (IHn y). intros k1 EQ1. rewrite EQ1. |
---|
| 997 | rewrite <- (Z_shift_add_bin_decomp x). |
---|
| 998 | rewrite ZBD. simpl. |
---|
| 999 | exists k1. |
---|
| 1000 | case b; unfold Z_shift_add; ring. |
---|
| 1001 | apply Z_of_bits_exten. intros. |
---|
| 1002 | rewrite zeq_false. decEq. omega. omega. |
---|
| 1003 | intro. exact (H wordsize x). |
---|
| 1004 | Qed. |
---|
| 1005 | |
---|
| 1006 | Lemma bits_of_Z_zero: |
---|
| 1007 | forall n x, bits_of_Z n 0 x = false. |
---|
| 1008 | Proof. |
---|
| 1009 | induction n; simpl; intros. |
---|
| 1010 | auto. |
---|
| 1011 | case (zeq x 0); intro. auto. auto. |
---|
| 1012 | Qed. |
---|
| 1013 | |
---|
| 1014 | Remark Z_bin_decomp_2xm1: |
---|
| 1015 | forall x, Z_bin_decomp (2 * x - 1) = (true, x - 1). |
---|
| 1016 | Proof. |
---|
| 1017 | intros. caseEq (Z_bin_decomp (2 * x - 1)). intros b y EQ. |
---|
| 1018 | generalize (Z_shift_add_bin_decomp (2 * x - 1)). |
---|
| 1019 | rewrite EQ; simpl. |
---|
| 1020 | replace (2 * x - 1) with (Z_shift_add true (x - 1)). |
---|
| 1021 | intro. elim (Z_shift_add_inj _ _ _ _ H); intros. |
---|
| 1022 | congruence. unfold Z_shift_add. omega. |
---|
| 1023 | Qed. |
---|
| 1024 | |
---|
| 1025 | Lemma bits_of_Z_mone: |
---|
| 1026 | forall n x, |
---|
| 1027 | 0 <= x < Z_of_nat n -> |
---|
| 1028 | bits_of_Z n (two_power_nat n - 1) x = true. |
---|
| 1029 | Proof. |
---|
| 1030 | induction n; intros. |
---|
| 1031 | simpl in H. omegaContradiction. |
---|
| 1032 | unfold bits_of_Z; fold bits_of_Z. |
---|
| 1033 | rewrite two_power_nat_S. rewrite Z_bin_decomp_2xm1. |
---|
| 1034 | rewrite inj_S in H. case (zeq x 0); intro. auto. |
---|
| 1035 | apply IHn. omega. |
---|
| 1036 | Qed. |
---|
| 1037 | |
---|
| 1038 | Lemma Z_bin_decomp_shift_add: |
---|
| 1039 | forall b x, Z_bin_decomp (Z_shift_add b x) = (b, x). |
---|
| 1040 | Proof. |
---|
| 1041 | intros. caseEq (Z_bin_decomp (Z_shift_add b x)); intros b' x' EQ. |
---|
| 1042 | generalize (Z_shift_add_bin_decomp (Z_shift_add b x)). |
---|
| 1043 | rewrite EQ; simpl fst; simpl snd. intro. |
---|
| 1044 | elim (Z_shift_add_inj _ _ _ _ H); intros. |
---|
| 1045 | congruence. |
---|
| 1046 | Qed. |
---|
| 1047 | |
---|
| 1048 | Lemma bits_of_Z_of_bits: |
---|
| 1049 | forall n f i, |
---|
| 1050 | 0 <= i < Z_of_nat n -> |
---|
| 1051 | bits_of_Z n (Z_of_bits n f) i = f i. |
---|
| 1052 | Proof. |
---|
| 1053 | induction n; intros; simpl. |
---|
| 1054 | simpl in H. omegaContradiction. |
---|
| 1055 | rewrite Z_bin_decomp_shift_add. |
---|
| 1056 | case (zeq i 0); intro. |
---|
| 1057 | congruence. |
---|
| 1058 | rewrite IHn. decEq. omega. rewrite inj_S in H. omega. |
---|
| 1059 | Qed. |
---|
| 1060 | |
---|
| 1061 | Lemma Z_of_bits_range: |
---|
| 1062 | forall f, 0 <= Z_of_bits wordsize f < modulus. |
---|
| 1063 | Proof. |
---|
| 1064 | unfold max_unsigned, modulus. |
---|
| 1065 | generalize wordsize. induction n; simpl; intros. |
---|
| 1066 | rewrite two_power_nat_O. omega. |
---|
| 1067 | rewrite two_power_nat_S. generalize (IHn (fun i => f (i + 1))). |
---|
| 1068 | set (x := Z_of_bits n (fun i => f (i + 1))). |
---|
| 1069 | intro. destruct (f 0); unfold Z_shift_add; omega. |
---|
| 1070 | Qed. |
---|
| 1071 | Hint Resolve Z_of_bits_range: ints. |
---|
| 1072 | |
---|
| 1073 | Lemma Z_of_bits_range_2: |
---|
| 1074 | forall f, 0 <= Z_of_bits wordsize f <= max_unsigned. |
---|
| 1075 | Proof. |
---|
| 1076 | intros. unfold max_unsigned. |
---|
| 1077 | generalize (Z_of_bits_range f). omega. |
---|
| 1078 | Qed. |
---|
| 1079 | Hint Resolve Z_of_bits_range_2: ints. |
---|
| 1080 | |
---|
| 1081 | Lemma bits_of_Z_below: |
---|
| 1082 | forall n x i, i < 0 -> bits_of_Z n x i = false. |
---|
| 1083 | Proof. |
---|
| 1084 | induction n; simpl; intros. |
---|
| 1085 | reflexivity. |
---|
| 1086 | destruct (Z_bin_decomp x). rewrite zeq_false. apply IHn. |
---|
| 1087 | omega. omega. |
---|
| 1088 | Qed. |
---|
| 1089 | |
---|
| 1090 | Lemma bits_of_Z_above: |
---|
| 1091 | forall n x i, i >= Z_of_nat n -> bits_of_Z n x i = false. |
---|
| 1092 | Proof. |
---|
| 1093 | induction n; intros; simpl. |
---|
| 1094 | reflexivity. |
---|
| 1095 | destruct (Z_bin_decomp x). rewrite zeq_false. apply IHn. |
---|
| 1096 | rewrite inj_S in H. omega. rewrite inj_S in H. omega. |
---|
| 1097 | Qed. |
---|
| 1098 | |
---|
| 1099 | Lemma bits_of_Z_of_bits': |
---|
| 1100 | forall n f i, |
---|
| 1101 | bits_of_Z n (Z_of_bits n f) i = |
---|
| 1102 | if zlt i 0 then false |
---|
| 1103 | else if zle (Z_of_nat n) i then false |
---|
| 1104 | else f i. |
---|
| 1105 | Proof. |
---|
| 1106 | intros. |
---|
| 1107 | destruct (zlt i 0). apply bits_of_Z_below; auto. |
---|
| 1108 | destruct (zle (Z_of_nat n) i). apply bits_of_Z_above. omega. |
---|
| 1109 | apply bits_of_Z_of_bits. omega. |
---|
| 1110 | Qed. |
---|
| 1111 | |
---|
| 1112 | Opaque Zmult. |
---|
| 1113 | |
---|
| 1114 | Lemma Z_of_bits_excl: |
---|
| 1115 | forall n f g h, |
---|
| 1116 | (forall i, 0 <= i < Z_of_nat n -> f i && g i = false) -> |
---|
| 1117 | (forall i, 0 <= i < Z_of_nat n -> f i || g i = h i) -> |
---|
| 1118 | Z_of_bits n f + Z_of_bits n g = Z_of_bits n h. |
---|
| 1119 | Proof. |
---|
| 1120 | induction n. |
---|
| 1121 | intros; reflexivity. |
---|
| 1122 | intros. simpl. rewrite inj_S in H. rewrite inj_S in H0. |
---|
| 1123 | rewrite <- (IHn (fun i => f(i+1)) (fun i => g(i+1)) (fun i => h(i+1))). |
---|
| 1124 | assert (0 <= 0 < Zsucc(Z_of_nat n)). omega. |
---|
| 1125 | unfold Z_shift_add. |
---|
| 1126 | rewrite <- H0; auto. |
---|
| 1127 | set (F := Z_of_bits n (fun i => f(i + 1))). |
---|
| 1128 | set (G := Z_of_bits n (fun i => g(i + 1))). |
---|
| 1129 | caseEq (f 0); intros; caseEq (g 0); intros; simpl. |
---|
| 1130 | generalize (H 0 H1). rewrite H2; rewrite H3. simpl. intros; discriminate. |
---|
| 1131 | omega. omega. omega. |
---|
| 1132 | intros; apply H. omega. |
---|
| 1133 | intros; apply H0. omega. |
---|
| 1134 | Qed. |
---|
| 1135 | |
---|
| 1136 | (** ** Properties of bitwise and, or, xor *) |
---|
| 1137 | |
---|
| 1138 | Lemma bitwise_binop_commut: |
---|
| 1139 | forall f, |
---|
| 1140 | (forall a b, f a b = f b a) -> |
---|
| 1141 | forall x y, |
---|
| 1142 | bitwise_binop f x y = bitwise_binop f y x. |
---|
| 1143 | Proof. |
---|
| 1144 | unfold bitwise_binop; intros. |
---|
| 1145 | decEq. apply Z_of_bits_exten; intros. auto. |
---|
| 1146 | Qed. |
---|
| 1147 | |
---|
| 1148 | Lemma bitwise_binop_assoc: |
---|
| 1149 | forall f, |
---|
| 1150 | (forall a b c, f a (f b c) = f (f a b) c) -> |
---|
| 1151 | forall x y z, |
---|
| 1152 | bitwise_binop f (bitwise_binop f x y) z = |
---|
| 1153 | bitwise_binop f x (bitwise_binop f y z). |
---|
| 1154 | Proof. |
---|
| 1155 | unfold bitwise_binop; intros. |
---|
| 1156 | repeat rewrite unsigned_repr; auto with ints. |
---|
| 1157 | decEq. apply Z_of_bits_exten; intros. |
---|
| 1158 | repeat (rewrite bits_of_Z_of_bits; auto). |
---|
| 1159 | Qed. |
---|
| 1160 | |
---|
| 1161 | Lemma bitwise_binop_idem: |
---|
| 1162 | forall f, |
---|
| 1163 | (forall a, f a a = a) -> |
---|
| 1164 | forall x, |
---|
| 1165 | bitwise_binop f x x = x. |
---|
| 1166 | Proof. |
---|
| 1167 | unfold bitwise_binop; intros. |
---|
| 1168 | transitivity (repr (Z_of_bits wordsize (bits_of_Z wordsize (unsigned x)))). |
---|
| 1169 | decEq. apply Z_of_bits_exten; intros. auto. |
---|
| 1170 | transitivity (repr (unsigned x)). |
---|
| 1171 | apply eqm_samerepr. apply Z_of_bits_of_Z. apply repr_unsigned. |
---|
| 1172 | Qed. |
---|
| 1173 | |
---|
| 1174 | Theorem and_commut: forall x y, and x y = and y x. |
---|
| 1175 | Proof (bitwise_binop_commut andb andb_comm). |
---|
| 1176 | |
---|
| 1177 | Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z). |
---|
| 1178 | Proof (bitwise_binop_assoc andb andb_assoc). |
---|
| 1179 | |
---|
| 1180 | Theorem and_zero: forall x, and x zero = zero. |
---|
| 1181 | Proof. |
---|
| 1182 | intros. unfold and, bitwise_binop. |
---|
| 1183 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1184 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1185 | rewrite unsigned_zero. rewrite bits_of_Z_zero. apply andb_b_false. |
---|
| 1186 | Qed. |
---|
| 1187 | |
---|
| 1188 | Theorem and_mone: forall x, and x mone = x. |
---|
| 1189 | Proof. |
---|
| 1190 | intros. unfold and, bitwise_binop. |
---|
| 1191 | transitivity (repr(unsigned x)). |
---|
| 1192 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1193 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1194 | rewrite unsigned_mone. rewrite bits_of_Z_mone. apply andb_b_true. auto. |
---|
| 1195 | apply repr_unsigned. |
---|
| 1196 | Qed. |
---|
| 1197 | |
---|
| 1198 | Theorem and_idem: forall x, and x x = x. |
---|
| 1199 | Proof. |
---|
| 1200 | assert (forall b, b && b = b). |
---|
| 1201 | destruct b; reflexivity. |
---|
| 1202 | exact (bitwise_binop_idem andb H). |
---|
| 1203 | Qed. |
---|
| 1204 | |
---|
| 1205 | Theorem or_commut: forall x y, or x y = or y x. |
---|
| 1206 | Proof (bitwise_binop_commut orb orb_comm). |
---|
| 1207 | |
---|
| 1208 | Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z). |
---|
| 1209 | Proof (bitwise_binop_assoc orb orb_assoc). |
---|
| 1210 | |
---|
| 1211 | Theorem or_zero: forall x, or x zero = x. |
---|
| 1212 | Proof. |
---|
| 1213 | intros. unfold or, bitwise_binop. |
---|
| 1214 | transitivity (repr(unsigned x)). |
---|
| 1215 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1216 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1217 | rewrite unsigned_zero. rewrite bits_of_Z_zero. apply orb_b_false. |
---|
| 1218 | apply repr_unsigned. |
---|
| 1219 | Qed. |
---|
| 1220 | |
---|
| 1221 | Theorem or_mone: forall x, or x mone = mone. |
---|
| 1222 | Proof. |
---|
| 1223 | intros. unfold or, bitwise_binop. |
---|
| 1224 | transitivity (repr(unsigned mone)). |
---|
| 1225 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1226 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1227 | rewrite unsigned_mone. rewrite bits_of_Z_mone. apply orb_b_true. auto. |
---|
| 1228 | apply repr_unsigned. |
---|
| 1229 | Qed. |
---|
| 1230 | |
---|
| 1231 | Theorem or_idem: forall x, or x x = x. |
---|
| 1232 | Proof. |
---|
| 1233 | assert (forall b, b || b = b). |
---|
| 1234 | destruct b; reflexivity. |
---|
| 1235 | exact (bitwise_binop_idem orb H). |
---|
| 1236 | Qed. |
---|
| 1237 | |
---|
| 1238 | Theorem and_or_distrib: |
---|
| 1239 | forall x y z, |
---|
| 1240 | and x (or y z) = or (and x y) (and x z). |
---|
| 1241 | Proof. |
---|
| 1242 | intros; unfold and, or, bitwise_binop. |
---|
| 1243 | decEq. repeat rewrite unsigned_repr; auto with ints. |
---|
| 1244 | apply Z_of_bits_exten; intros. |
---|
| 1245 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1246 | apply demorgan1. |
---|
| 1247 | Qed. |
---|
| 1248 | |
---|
| 1249 | Theorem xor_commut: forall x y, xor x y = xor y x. |
---|
| 1250 | Proof (bitwise_binop_commut xorb xorb_comm). |
---|
| 1251 | |
---|
| 1252 | Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z). |
---|
| 1253 | Proof. |
---|
| 1254 | assert (forall a b c, xorb a (xorb b c) = xorb (xorb a b) c). |
---|
| 1255 | symmetry. apply xorb_assoc. |
---|
| 1256 | exact (bitwise_binop_assoc xorb H). |
---|
| 1257 | Qed. |
---|
| 1258 | |
---|
| 1259 | Theorem xor_zero: forall x, xor x zero = x. |
---|
| 1260 | Proof. |
---|
| 1261 | intros. unfold xor, bitwise_binop. |
---|
| 1262 | transitivity (repr(unsigned x)). |
---|
| 1263 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1264 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1265 | rewrite unsigned_zero. rewrite bits_of_Z_zero. apply xorb_false. |
---|
| 1266 | apply repr_unsigned. |
---|
| 1267 | Qed. |
---|
| 1268 | |
---|
| 1269 | Theorem xor_idem: forall x, xor x x = zero. |
---|
| 1270 | Proof. |
---|
| 1271 | intros. unfold xor, bitwise_binop. |
---|
| 1272 | transitivity (repr(unsigned zero)). |
---|
| 1273 | apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1274 | apply eqm_refl2. apply Z_of_bits_exten. intros. |
---|
| 1275 | rewrite unsigned_zero. rewrite bits_of_Z_zero. apply xorb_nilpotent. |
---|
| 1276 | apply repr_unsigned. |
---|
| 1277 | Qed. |
---|
| 1278 | |
---|
| 1279 | Theorem xor_zero_one: xor zero one = one. |
---|
| 1280 | Proof. rewrite xor_commut. apply xor_zero. Qed. |
---|
| 1281 | |
---|
| 1282 | Theorem xor_one_one: xor one one = zero. |
---|
| 1283 | Proof. apply xor_idem. Qed. |
---|
| 1284 | |
---|
| 1285 | Theorem and_xor_distrib: |
---|
| 1286 | forall x y z, |
---|
| 1287 | and x (xor y z) = xor (and x y) (and x z). |
---|
| 1288 | Proof. |
---|
| 1289 | intros; unfold and, xor, bitwise_binop. |
---|
| 1290 | decEq. repeat rewrite unsigned_repr; auto with ints. |
---|
| 1291 | apply Z_of_bits_exten; intros. |
---|
| 1292 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1293 | assert (forall a b c, a && (xorb b c) = xorb (a && b) (a && c)). |
---|
| 1294 | destruct a; destruct b; destruct c; reflexivity. |
---|
| 1295 | auto. |
---|
| 1296 | Qed. |
---|
| 1297 | |
---|
| 1298 | Theorem not_involutive: |
---|
| 1299 | forall (x: int), not (not x) = x. |
---|
| 1300 | Proof. |
---|
| 1301 | intros. unfold not. rewrite xor_assoc. rewrite xor_idem. apply xor_zero. |
---|
| 1302 | Qed. |
---|
| 1303 | |
---|
| 1304 | (** ** Properties of shifts and rotates *) |
---|
| 1305 | |
---|
| 1306 | Lemma Z_of_bits_shift: |
---|
| 1307 | forall n f, |
---|
| 1308 | exists k, |
---|
| 1309 | Z_of_bits n (fun i => f (i - 1)) = |
---|
| 1310 | k * two_power_nat n + Z_shift_add (f (-1)) (Z_of_bits n f). |
---|
| 1311 | Proof. |
---|
| 1312 | induction n; intros. |
---|
| 1313 | simpl. rewrite two_power_nat_O. unfold Z_shift_add. |
---|
| 1314 | exists (if f (-1) then (-1) else 0). |
---|
| 1315 | destruct (f (-1)); omega. |
---|
| 1316 | rewrite two_power_nat_S. simpl. |
---|
| 1317 | elim (IHn (fun i => f (i + 1))). intros k' EQ. |
---|
| 1318 | replace (Z_of_bits n (fun i => f (i - 1 + 1))) |
---|
| 1319 | with (Z_of_bits n (fun i => f (i + 1 - 1))) in EQ. |
---|
| 1320 | rewrite EQ. |
---|
| 1321 | change (-1 + 1) with 0. |
---|
| 1322 | exists k'. |
---|
| 1323 | unfold Z_shift_add; destruct (f (-1)); destruct (f 0); ring. |
---|
| 1324 | apply Z_of_bits_exten; intros. |
---|
| 1325 | decEq. omega. |
---|
| 1326 | Qed. |
---|
| 1327 | |
---|
| 1328 | Lemma Z_of_bits_shifts: |
---|
| 1329 | forall m f, |
---|
| 1330 | 0 <= m -> |
---|
| 1331 | (forall i, i < 0 -> f i = false) -> |
---|
| 1332 | eqm (Z_of_bits wordsize (fun i => f (i - m))) |
---|
| 1333 | (two_p m * Z_of_bits wordsize f). |
---|
| 1334 | Proof. |
---|
| 1335 | intros. pattern m. apply natlike_ind. |
---|
| 1336 | apply eqm_refl2. transitivity (Z_of_bits wordsize f). |
---|
| 1337 | apply Z_of_bits_exten; intros. decEq. omega. |
---|
| 1338 | simpl two_p. omega. |
---|
| 1339 | intros. rewrite two_p_S; auto. |
---|
| 1340 | set (f' := fun i => f (i - x)). |
---|
| 1341 | apply eqm_trans with (Z_of_bits wordsize (fun i => f' (i - 1))). |
---|
| 1342 | apply eqm_refl2. apply Z_of_bits_exten; intros. |
---|
| 1343 | unfold f'. decEq. omega. |
---|
| 1344 | apply eqm_trans with (Z_shift_add (f' (-1)) (Z_of_bits wordsize f')). |
---|
| 1345 | exact (Z_of_bits_shift wordsize f'). |
---|
| 1346 | unfold f'. unfold Z_shift_add. rewrite H0. |
---|
| 1347 | rewrite <- Zmult_assoc. apply eqm_mult. apply eqm_refl. |
---|
| 1348 | apply H2. omega. assumption. |
---|
| 1349 | Qed. |
---|
| 1350 | |
---|
| 1351 | Lemma shl_mul_two_p: |
---|
| 1352 | forall x y, |
---|
| 1353 | shl x y = mul x (repr (two_p (unsigned y))). |
---|
| 1354 | Proof. |
---|
| 1355 | intros. unfold shl, mul. |
---|
| 1356 | apply eqm_samerepr. |
---|
| 1357 | eapply eqm_trans. |
---|
| 1358 | apply Z_of_bits_shifts. |
---|
| 1359 | generalize (unsigned_range y). omega. |
---|
| 1360 | intros; apply bits_of_Z_below; auto. |
---|
| 1361 | rewrite Zmult_comm. apply eqm_mult. |
---|
| 1362 | apply Z_of_bits_of_Z. apply eqm_unsigned_repr. |
---|
| 1363 | Qed. |
---|
| 1364 | |
---|
| 1365 | Theorem shl_zero: forall x, shl x zero = x. |
---|
| 1366 | Proof. |
---|
| 1367 | intros. rewrite shl_mul_two_p. |
---|
| 1368 | change (repr (two_p (unsigned zero))) with one. |
---|
| 1369 | apply mul_one. |
---|
| 1370 | Qed. |
---|
| 1371 | |
---|
| 1372 | Theorem shl_mul: |
---|
| 1373 | forall x y, |
---|
| 1374 | shl x y = mul x (shl one y). |
---|
| 1375 | Proof. |
---|
| 1376 | intros. |
---|
| 1377 | assert (shl one y = repr (two_p (unsigned y))). |
---|
| 1378 | rewrite shl_mul_two_p. rewrite mul_commut. rewrite mul_one. auto. |
---|
| 1379 | rewrite H. apply shl_mul_two_p. |
---|
| 1380 | Qed. |
---|
| 1381 | |
---|
| 1382 | Lemma ltu_inv: |
---|
| 1383 | forall x y, ltu x y = true -> 0 <= unsigned x < unsigned y. |
---|
| 1384 | Proof. |
---|
| 1385 | unfold ltu; intros. destruct (zlt (unsigned x) (unsigned y)). |
---|
| 1386 | split; auto. generalize (unsigned_range x); omega. |
---|
| 1387 | discriminate. |
---|
| 1388 | Qed. |
---|
| 1389 | |
---|
| 1390 | Theorem shl_rolm: |
---|
| 1391 | forall x n, |
---|
| 1392 | ltu n iwordsize = true -> |
---|
| 1393 | shl x n = rolm x n (shl mone n). |
---|
| 1394 | Proof. |
---|
| 1395 | intros. exploit ltu_inv; eauto. rewrite unsigned_repr_wordsize; intros. |
---|
| 1396 | unfold shl, rolm, rol, and, bitwise_binop. |
---|
| 1397 | decEq. apply Z_of_bits_exten; intros. |
---|
| 1398 | repeat rewrite unsigned_repr; auto with ints. |
---|
| 1399 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1400 | case (zlt z (unsigned n)); intro LT2. |
---|
| 1401 | assert (z - unsigned n < 0). omega. |
---|
| 1402 | rewrite (bits_of_Z_below wordsize (unsigned x) _ H2). |
---|
| 1403 | rewrite (bits_of_Z_below wordsize (unsigned mone) _ H2). |
---|
| 1404 | symmetry. apply andb_b_false. |
---|
| 1405 | assert (z - unsigned n < Z_of_nat wordsize). |
---|
| 1406 | generalize (unsigned_range n). omega. |
---|
| 1407 | rewrite unsigned_mone. |
---|
| 1408 | rewrite bits_of_Z_mone. rewrite andb_b_true. decEq. |
---|
| 1409 | rewrite Zmod_small. auto. omega. omega. |
---|
| 1410 | Qed. |
---|
| 1411 | |
---|
| 1412 | Lemma bitwise_binop_shl: |
---|
| 1413 | forall f x y n, |
---|
| 1414 | f false false = false -> |
---|
| 1415 | bitwise_binop f (shl x n) (shl y n) = shl (bitwise_binop f x y) n. |
---|
| 1416 | Proof. |
---|
| 1417 | intros. unfold bitwise_binop, shl. |
---|
| 1418 | decEq. repeat rewrite unsigned_repr; auto with ints. |
---|
| 1419 | apply Z_of_bits_exten; intros. |
---|
| 1420 | case (zlt (z - unsigned n) 0); intro. |
---|
| 1421 | transitivity false. repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1422 | repeat rewrite bits_of_Z_below; auto. |
---|
| 1423 | rewrite bits_of_Z_below; auto. |
---|
| 1424 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1425 | generalize (unsigned_range n). omega. |
---|
| 1426 | Qed. |
---|
| 1427 | |
---|
| 1428 | Theorem and_shl: |
---|
| 1429 | forall x y n, |
---|
| 1430 | and (shl x n) (shl y n) = shl (and x y) n. |
---|
| 1431 | Proof. |
---|
| 1432 | unfold and; intros. apply bitwise_binop_shl. reflexivity. |
---|
| 1433 | Qed. |
---|
| 1434 | |
---|
| 1435 | |
---|
| 1436 | Theorem shl_shl: |
---|
| 1437 | forall x y z, |
---|
| 1438 | ltu y iwordsize = true -> |
---|
| 1439 | ltu z iwordsize = true -> |
---|
| 1440 | ltu (add y z) iwordsize = true -> |
---|
| 1441 | shl (shl x y) z = shl x (add y z). |
---|
| 1442 | Proof. |
---|
| 1443 | intros. unfold shl, add. |
---|
| 1444 | generalize (ltu_inv _ _ H). |
---|
| 1445 | generalize (ltu_inv _ _ H0). |
---|
| 1446 | rewrite unsigned_repr_wordsize. |
---|
| 1447 | set (x' := unsigned x). |
---|
| 1448 | set (y' := unsigned y). |
---|
| 1449 | set (z' := unsigned z). |
---|
| 1450 | intros. |
---|
| 1451 | repeat rewrite unsigned_repr. |
---|
| 1452 | decEq. apply Z_of_bits_exten. intros n R. |
---|
| 1453 | rewrite bits_of_Z_of_bits'. |
---|
| 1454 | destruct (zlt (n - z') 0). |
---|
| 1455 | symmetry. apply bits_of_Z_below. omega. |
---|
| 1456 | destruct (zle (Z_of_nat wordsize) (n - z')). |
---|
| 1457 | symmetry. apply bits_of_Z_below. omega. |
---|
| 1458 | decEq. omega. |
---|
| 1459 | generalize two_wordsize_max_unsigned; omega. |
---|
| 1460 | apply Z_of_bits_range_2. |
---|
| 1461 | Qed. |
---|
| 1462 | |
---|
| 1463 | Theorem shru_shru: |
---|
| 1464 | forall x y z, |
---|
| 1465 | ltu y iwordsize = true -> |
---|
| 1466 | ltu z iwordsize = true -> |
---|
| 1467 | ltu (add y z) iwordsize = true -> |
---|
| 1468 | shru (shru x y) z = shru x (add y z). |
---|
| 1469 | Proof. |
---|
| 1470 | intros. unfold shru, add. |
---|
| 1471 | generalize (ltu_inv _ _ H). |
---|
| 1472 | generalize (ltu_inv _ _ H0). |
---|
| 1473 | rewrite unsigned_repr_wordsize. |
---|
| 1474 | set (x' := unsigned x). |
---|
| 1475 | set (y' := unsigned y). |
---|
| 1476 | set (z' := unsigned z). |
---|
| 1477 | intros. |
---|
| 1478 | repeat rewrite unsigned_repr. |
---|
| 1479 | decEq. apply Z_of_bits_exten. intros n R. |
---|
| 1480 | rewrite bits_of_Z_of_bits'. |
---|
| 1481 | destruct (zlt (n + z') 0). omegaContradiction. |
---|
| 1482 | destruct (zle (Z_of_nat wordsize) (n + z')). |
---|
| 1483 | symmetry. apply bits_of_Z_above. omega. |
---|
| 1484 | decEq. omega. |
---|
| 1485 | generalize two_wordsize_max_unsigned; omega. |
---|
| 1486 | apply Z_of_bits_range_2. |
---|
| 1487 | Qed. |
---|
| 1488 | |
---|
| 1489 | Theorem shru_rolm: |
---|
| 1490 | forall x n, |
---|
| 1491 | ltu n iwordsize = true -> |
---|
| 1492 | shru x n = rolm x (sub iwordsize n) (shru mone n). |
---|
| 1493 | Proof. |
---|
| 1494 | intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize. intro. |
---|
| 1495 | unfold shru, rolm, rol, and, bitwise_binop. |
---|
| 1496 | decEq. apply Z_of_bits_exten; intros. |
---|
| 1497 | repeat rewrite unsigned_repr; auto with ints. |
---|
| 1498 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1499 | unfold sub. rewrite unsigned_repr_wordsize. |
---|
| 1500 | rewrite unsigned_repr. |
---|
| 1501 | case (zlt (z + unsigned n) (Z_of_nat wordsize)); intro LT2. |
---|
| 1502 | rewrite unsigned_mone. rewrite bits_of_Z_mone. rewrite andb_b_true. |
---|
| 1503 | decEq. |
---|
| 1504 | replace (z - (Z_of_nat wordsize - unsigned n)) |
---|
| 1505 | with ((z + unsigned n) + (-1) * Z_of_nat wordsize). |
---|
| 1506 | rewrite Z_mod_plus. symmetry. apply Zmod_small. |
---|
| 1507 | generalize (unsigned_range n). omega. omega. omega. |
---|
| 1508 | generalize (unsigned_range n). omega. |
---|
| 1509 | rewrite (bits_of_Z_above wordsize (unsigned x) _ LT2). |
---|
| 1510 | rewrite (bits_of_Z_above wordsize (unsigned mone) _ LT2). |
---|
| 1511 | symmetry. apply andb_b_false. |
---|
| 1512 | split. omega. apply Zle_trans with (Z_of_nat wordsize). |
---|
| 1513 | generalize (unsigned_range n); omega. apply wordsize_max_unsigned. |
---|
| 1514 | Qed. |
---|
| 1515 | |
---|
| 1516 | Lemma bitwise_binop_shru: |
---|
| 1517 | forall f x y n, |
---|
| 1518 | f false false = false -> |
---|
| 1519 | bitwise_binop f (shru x n) (shru y n) = shru (bitwise_binop f x y) n. |
---|
| 1520 | Proof. |
---|
| 1521 | intros. unfold bitwise_binop, shru. |
---|
| 1522 | decEq. repeat rewrite unsigned_repr; auto with ints. |
---|
| 1523 | apply Z_of_bits_exten; intros. |
---|
| 1524 | case (zlt (z + unsigned n) (Z_of_nat wordsize)); intro. |
---|
| 1525 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1526 | generalize (unsigned_range n); omega. |
---|
| 1527 | transitivity false. repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1528 | repeat rewrite bits_of_Z_above; auto. |
---|
| 1529 | rewrite bits_of_Z_above; auto. |
---|
| 1530 | Qed. |
---|
| 1531 | |
---|
| 1532 | Lemma and_shru: |
---|
| 1533 | forall x y n, |
---|
| 1534 | and (shru x n) (shru y n) = shru (and x y) n. |
---|
| 1535 | Proof. |
---|
| 1536 | unfold and; intros. apply bitwise_binop_shru. reflexivity. |
---|
| 1537 | Qed. |
---|
| 1538 | |
---|
| 1539 | Theorem shr_shr: |
---|
| 1540 | forall x y z, |
---|
| 1541 | ltu y iwordsize = true -> |
---|
| 1542 | ltu z iwordsize = true -> |
---|
| 1543 | ltu (add y z) iwordsize = true -> |
---|
| 1544 | shr (shr x y) z = shr x (add y z). |
---|
| 1545 | Proof. |
---|
| 1546 | intros. unfold shr, add. |
---|
| 1547 | generalize (ltu_inv _ _ H). |
---|
| 1548 | generalize (ltu_inv _ _ H0). |
---|
| 1549 | rewrite unsigned_repr_wordsize. |
---|
| 1550 | set (x' := signed x). |
---|
| 1551 | set (y' := unsigned y). |
---|
| 1552 | set (z' := unsigned z). |
---|
| 1553 | intros. |
---|
| 1554 | rewrite unsigned_repr. |
---|
| 1555 | rewrite two_p_is_exp. |
---|
| 1556 | rewrite signed_repr. |
---|
| 1557 | decEq. apply Zdiv_Zdiv. apply two_p_gt_ZERO. omega. apply two_p_gt_ZERO. omega. |
---|
| 1558 | apply Zdiv_interval_2. unfold x'; apply signed_range. |
---|
| 1559 | generalize min_signed_neg; omega. |
---|
| 1560 | generalize max_signed_pos; omega. |
---|
| 1561 | apply two_p_gt_ZERO. omega. omega. omega. |
---|
| 1562 | generalize two_wordsize_max_unsigned; omega. |
---|
| 1563 | Qed. |
---|
| 1564 | |
---|
| 1565 | Theorem rol_zero: |
---|
| 1566 | forall x, |
---|
| 1567 | rol x zero = x. |
---|
| 1568 | Proof. |
---|
| 1569 | intros. transitivity (repr (unsigned x)). |
---|
| 1570 | unfold rol. apply eqm_samerepr. eapply eqm_trans. 2: apply Z_of_bits_of_Z. |
---|
| 1571 | apply eqm_refl2. apply Z_of_bits_exten; intros. decEq. rewrite unsigned_zero. |
---|
| 1572 | replace (z - 0) with z by omega. apply Zmod_small. auto. |
---|
| 1573 | apply repr_unsigned. |
---|
| 1574 | Qed. |
---|
| 1575 | |
---|
| 1576 | Lemma bitwise_binop_rol: |
---|
| 1577 | forall f x y n, |
---|
| 1578 | bitwise_binop f (rol x n) (rol y n) = rol (bitwise_binop f x y) n. |
---|
| 1579 | Proof. |
---|
| 1580 | intros. unfold bitwise_binop, rol. |
---|
| 1581 | decEq. repeat (rewrite unsigned_repr; auto with ints). |
---|
| 1582 | apply Z_of_bits_exten; intros. |
---|
| 1583 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1584 | apply Z_mod_lt. generalize wordsize_pos; omega. |
---|
| 1585 | Qed. |
---|
| 1586 | |
---|
| 1587 | Theorem rol_and: |
---|
| 1588 | forall x y n, |
---|
| 1589 | rol (and x y) n = and (rol x n) (rol y n). |
---|
| 1590 | Proof. |
---|
| 1591 | intros. symmetry. unfold and. apply bitwise_binop_rol. |
---|
| 1592 | Qed. |
---|
| 1593 | |
---|
| 1594 | Theorem rol_rol: |
---|
| 1595 | forall x n m, |
---|
| 1596 | Zdivide (Z_of_nat wordsize) modulus -> |
---|
| 1597 | rol (rol x n) m = rol x (modu (add n m) iwordsize). |
---|
| 1598 | Proof. |
---|
| 1599 | intros. unfold rol. decEq. |
---|
| 1600 | repeat (rewrite unsigned_repr; auto with ints). |
---|
| 1601 | apply Z_of_bits_exten; intros. |
---|
| 1602 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1603 | decEq. unfold modu, add. |
---|
| 1604 | set (W := Z_of_nat wordsize). |
---|
| 1605 | set (M := unsigned m); set (N := unsigned n). |
---|
| 1606 | assert (W > 0). unfold W; generalize wordsize_pos; omega. |
---|
| 1607 | assert (forall a, eqmod W a (unsigned (repr a))). |
---|
| 1608 | intros. eapply eqmod_divides. apply eqm_unsigned_repr. assumption. |
---|
| 1609 | apply eqmod_mod_eq. auto. |
---|
| 1610 | replace (unsigned iwordsize) with W. |
---|
| 1611 | apply eqmod_trans with (z - (N + M) mod W). |
---|
| 1612 | apply eqmod_trans with ((z - M) - N). |
---|
| 1613 | apply eqmod_sub. apply eqmod_sym. apply eqmod_mod. auto. |
---|
| 1614 | apply eqmod_refl. |
---|
| 1615 | replace (z - M - N) with (z - (N + M)). |
---|
| 1616 | apply eqmod_sub. apply eqmod_refl. apply eqmod_mod. auto. |
---|
| 1617 | omega. |
---|
| 1618 | apply eqmod_sub. apply eqmod_refl. |
---|
| 1619 | eapply eqmod_trans; [idtac|apply H2]. |
---|
| 1620 | eapply eqmod_trans; [idtac|apply eqmod_mod]. |
---|
| 1621 | apply eqmod_sym. eapply eqmod_trans; [idtac|apply eqmod_mod]. |
---|
| 1622 | apply eqmod_sym. apply H2. auto. auto. |
---|
| 1623 | symmetry. unfold W. apply unsigned_repr_wordsize. |
---|
| 1624 | apply Z_mod_lt. generalize wordsize_pos; omega. |
---|
| 1625 | Qed. |
---|
| 1626 | |
---|
| 1627 | Theorem rolm_zero: |
---|
| 1628 | forall x m, |
---|
| 1629 | rolm x zero m = and x m. |
---|
| 1630 | Proof. |
---|
| 1631 | intros. unfold rolm. rewrite rol_zero. auto. |
---|
| 1632 | Qed. |
---|
| 1633 | |
---|
| 1634 | Theorem rolm_rolm: |
---|
| 1635 | forall x n1 m1 n2 m2, |
---|
| 1636 | Zdivide (Z_of_nat wordsize) modulus -> |
---|
| 1637 | rolm (rolm x n1 m1) n2 m2 = |
---|
| 1638 | rolm x (modu (add n1 n2) iwordsize) |
---|
| 1639 | (and (rol m1 n2) m2). |
---|
| 1640 | Proof. |
---|
| 1641 | intros. |
---|
| 1642 | unfold rolm. rewrite rol_and. rewrite and_assoc. |
---|
| 1643 | rewrite rol_rol. reflexivity. auto. |
---|
| 1644 | Qed. |
---|
| 1645 | |
---|
| 1646 | Theorem rol_or: |
---|
| 1647 | forall x y n, |
---|
| 1648 | rol (or x y) n = or (rol x n) (rol y n). |
---|
| 1649 | Proof. |
---|
| 1650 | intros. symmetry. unfold or. apply bitwise_binop_rol. |
---|
| 1651 | Qed. |
---|
| 1652 | |
---|
| 1653 | Theorem or_rolm: |
---|
| 1654 | forall x n m1 m2, |
---|
| 1655 | or (rolm x n m1) (rolm x n m2) = rolm x n (or m1 m2). |
---|
| 1656 | Proof. |
---|
| 1657 | intros; unfold rolm. symmetry. apply and_or_distrib. |
---|
| 1658 | Qed. |
---|
| 1659 | |
---|
| 1660 | Theorem ror_rol: |
---|
| 1661 | forall x y, |
---|
| 1662 | ltu y iwordsize = true -> |
---|
| 1663 | ror x y = rol x (sub iwordsize y). |
---|
| 1664 | Proof. |
---|
| 1665 | intros. unfold ror, rol, sub. |
---|
| 1666 | generalize (ltu_inv _ _ H). |
---|
| 1667 | rewrite unsigned_repr_wordsize. |
---|
| 1668 | intro. rewrite unsigned_repr. |
---|
| 1669 | decEq. apply Z_of_bits_exten. intros. decEq. |
---|
| 1670 | apply eqmod_mod_eq. omega. |
---|
| 1671 | exists 1. omega. |
---|
| 1672 | generalize wordsize_pos; generalize wordsize_max_unsigned; omega. |
---|
| 1673 | Qed. |
---|
| 1674 | |
---|
| 1675 | Theorem or_ror: |
---|
| 1676 | forall x y z, |
---|
| 1677 | ltu y iwordsize = true -> |
---|
| 1678 | ltu z iwordsize = true -> |
---|
| 1679 | add y z = iwordsize -> |
---|
| 1680 | ror x z = or (shl x y) (shru x z). |
---|
| 1681 | Proof. |
---|
| 1682 | intros. |
---|
| 1683 | generalize (ltu_inv _ _ H). |
---|
| 1684 | generalize (ltu_inv _ _ H0). |
---|
| 1685 | rewrite unsigned_repr_wordsize. |
---|
| 1686 | intros. |
---|
| 1687 | unfold or, bitwise_binop, shl, shru, ror. |
---|
| 1688 | set (ux := unsigned x). |
---|
| 1689 | decEq. apply Z_of_bits_exten. intros i iRANGE. |
---|
| 1690 | repeat rewrite unsigned_repr. |
---|
| 1691 | repeat rewrite bits_of_Z_of_bits; auto. |
---|
| 1692 | assert (y = sub iwordsize z). |
---|
| 1693 | rewrite <- H1. rewrite add_commut. rewrite sub_add_l. rewrite sub_idem. |
---|
| 1694 | rewrite add_commut. rewrite add_zero. auto. |
---|
| 1695 | assert (unsigned y = Z_of_nat wordsize - unsigned z). |
---|
| 1696 | rewrite H4. unfold sub. rewrite unsigned_repr_wordsize. apply unsigned_repr. |
---|
| 1697 | generalize wordsize_max_unsigned; omega. |
---|
| 1698 | destruct (zlt (i + unsigned z) (Z_of_nat wordsize)). |
---|
| 1699 | rewrite Zmod_small. |
---|
| 1700 | replace (bits_of_Z wordsize ux (i - unsigned y)) with false. |
---|
| 1701 | auto. |
---|
| 1702 | symmetry. apply bits_of_Z_below. omega. omega. |
---|
| 1703 | replace (bits_of_Z wordsize ux (i + unsigned z)) with false. |
---|
| 1704 | rewrite orb_false_r. decEq. |
---|
| 1705 | replace (i + unsigned z) with (i - unsigned y + 1 * Z_of_nat wordsize) by omega. |
---|
| 1706 | rewrite Z_mod_plus. apply Zmod_small. omega. generalize wordsize_pos; omega. |
---|
| 1707 | symmetry. apply bits_of_Z_above. auto. |
---|
| 1708 | apply Z_of_bits_range_2. apply Z_of_bits_range_2. |
---|
| 1709 | Qed. |
---|
| 1710 | |
---|
| 1711 | Lemma bits_of_Z_two_p: |
---|
| 1712 | forall n x i, |
---|
| 1713 | x >= 0 -> 0 <= i < Z_of_nat n -> |
---|
| 1714 | bits_of_Z n (two_p x - 1) i = zlt i x. |
---|
| 1715 | Proof. |
---|
| 1716 | induction n; intros. |
---|
| 1717 | simpl in H0. omegaContradiction. |
---|
| 1718 | destruct (zeq x 0). subst x. change (two_p 0 - 1) with 0. rewrite bits_of_Z_zero. |
---|
| 1719 | unfold proj_sumbool; rewrite zlt_false. auto. omega. |
---|
| 1720 | replace (two_p x) with (2 * two_p (x - 1)). simpl. rewrite Z_bin_decomp_2xm1. |
---|
| 1721 | destruct (zeq i 0). subst. unfold proj_sumbool. rewrite zlt_true. auto. omega. |
---|
| 1722 | rewrite inj_S in H0. rewrite IHn. unfold proj_sumbool. destruct (zlt i x). |
---|
| 1723 | apply zlt_true. omega. |
---|
| 1724 | apply zlt_false. omega. |
---|
| 1725 | omega. omega. rewrite <- two_p_S. decEq. omega. omega. |
---|
| 1726 | Qed. |
---|
| 1727 | |
---|
| 1728 | Remark two_p_m1_range: |
---|
| 1729 | forall n, |
---|
| 1730 | 0 <= n <= Z_of_nat wordsize -> |
---|
| 1731 | 0 <= two_p n - 1 <= max_unsigned. |
---|
| 1732 | Proof. |
---|
| 1733 | intros. split. |
---|
| 1734 | assert (two_p n > 0). apply two_p_gt_ZERO. omega. omega. |
---|
| 1735 | assert (two_p n <= two_p (Z_of_nat wordsize)). apply two_p_monotone. auto. |
---|
| 1736 | unfold max_unsigned. unfold modulus. rewrite two_power_nat_two_p. omega. |
---|
| 1737 | Qed. |
---|
| 1738 | |
---|
| 1739 | Theorem shru_shl_and: |
---|
| 1740 | forall x y, |
---|
| 1741 | ltu y iwordsize = true -> |
---|
| 1742 | shru (shl x y) y = and x (repr (two_p (Z_of_nat wordsize - unsigned y) - 1)). |
---|
| 1743 | Proof. |
---|
| 1744 | intros. exploit ltu_inv; eauto. rewrite unsigned_repr_wordsize. intros. |
---|
| 1745 | unfold and, bitwise_binop, shl, shru. |
---|
| 1746 | decEq. apply Z_of_bits_exten. intros. |
---|
| 1747 | repeat rewrite unsigned_repr. |
---|
| 1748 | rewrite bits_of_Z_two_p. |
---|
| 1749 | destruct (zlt (z + unsigned y) (Z_of_nat wordsize)). |
---|
| 1750 | rewrite bits_of_Z_of_bits. unfold proj_sumbool. rewrite zlt_true. |
---|
| 1751 | rewrite andb_true_r. f_equal. omega. |
---|
| 1752 | omega. omega. |
---|
| 1753 | rewrite bits_of_Z_above. unfold proj_sumbool. rewrite zlt_false. rewrite andb_false_r; auto. |
---|
| 1754 | omega. omega. omega. auto. |
---|
| 1755 | apply two_p_m1_range. omega. |
---|
| 1756 | apply Z_of_bits_range_2. |
---|
| 1757 | Qed. |
---|
| 1758 | |
---|
| 1759 | (** ** Relation between shifts and powers of 2 *) |
---|
| 1760 | |
---|
| 1761 | Fixpoint powerserie (l: list Z): Z := |
---|
| 1762 | match l with |
---|
| 1763 | | nil => 0 |
---|
| 1764 | | x :: xs => two_p x + powerserie xs |
---|
| 1765 | end. |
---|
| 1766 | |
---|
| 1767 | Lemma Z_bin_decomp_range: |
---|
| 1768 | forall x n, |
---|
| 1769 | 0 <= x < 2 * n -> 0 <= snd (Z_bin_decomp x) < n. |
---|
| 1770 | Proof. |
---|
| 1771 | intros. rewrite <- (Z_shift_add_bin_decomp x) in H. |
---|
| 1772 | unfold Z_shift_add in H. destruct (fst (Z_bin_decomp x)); omega. |
---|
| 1773 | Qed. |
---|
| 1774 | |
---|
| 1775 | Lemma Z_one_bits_powerserie: |
---|
| 1776 | forall x, 0 <= x < modulus -> x = powerserie (Z_one_bits wordsize x 0). |
---|
| 1777 | Proof. |
---|
| 1778 | assert (forall n x i, |
---|
| 1779 | 0 <= i -> |
---|
| 1780 | 0 <= x < two_power_nat n -> |
---|
| 1781 | x * two_p i = powerserie (Z_one_bits n x i)). |
---|
| 1782 | induction n; intros. |
---|
| 1783 | simpl. rewrite two_power_nat_O in H0. |
---|
| 1784 | assert (x = 0). omega. subst x. omega. |
---|
| 1785 | rewrite two_power_nat_S in H0. simpl Z_one_bits. |
---|
| 1786 | generalize (Z_shift_add_bin_decomp x). |
---|
| 1787 | generalize (Z_bin_decomp_range x _ H0). |
---|
| 1788 | case (Z_bin_decomp x). simpl. intros b y RANGE SHADD. |
---|
| 1789 | subst x. unfold Z_shift_add. |
---|
| 1790 | destruct b. simpl powerserie. rewrite <- IHn. |
---|
| 1791 | rewrite two_p_is_exp. change (two_p 1) with 2. ring. |
---|
| 1792 | auto. omega. omega. auto. |
---|
| 1793 | rewrite <- IHn. |
---|
| 1794 | rewrite two_p_is_exp. change (two_p 1) with 2. ring. |
---|
| 1795 | auto. omega. omega. auto. |
---|
| 1796 | intros. rewrite <- H. change (two_p 0) with 1. omega. |
---|
| 1797 | omega. exact H0. |
---|
| 1798 | Qed. |
---|
| 1799 | |
---|
| 1800 | Lemma Z_one_bits_range: |
---|
| 1801 | forall x i, In i (Z_one_bits wordsize x 0) -> 0 <= i < Z_of_nat wordsize. |
---|
| 1802 | Proof. |
---|
| 1803 | assert (forall n x i j, |
---|
| 1804 | In j (Z_one_bits n x i) -> i <= j < i + Z_of_nat n). |
---|
| 1805 | induction n; simpl In. |
---|
| 1806 | intros; elim H. |
---|
| 1807 | intros x i j. destruct (Z_bin_decomp x). case b. |
---|
| 1808 | rewrite inj_S. simpl. intros [A|B]. subst j. omega. |
---|
| 1809 | generalize (IHn _ _ _ B). omega. |
---|
| 1810 | intros B. rewrite inj_S. generalize (IHn _ _ _ B). omega. |
---|
| 1811 | intros. generalize (H wordsize x 0 i H0). omega. |
---|
| 1812 | Qed. |
---|
| 1813 | |
---|
| 1814 | Lemma is_power2_rng: |
---|
| 1815 | forall n logn, |
---|
| 1816 | is_power2 n = Some logn -> |
---|
| 1817 | 0 <= unsigned logn < Z_of_nat wordsize. |
---|
| 1818 | Proof. |
---|
| 1819 | intros n logn. unfold is_power2. |
---|
| 1820 | generalize (Z_one_bits_range (unsigned n)). |
---|
| 1821 | destruct (Z_one_bits wordsize (unsigned n) 0). |
---|
| 1822 | intros; discriminate. |
---|
| 1823 | destruct l. |
---|
| 1824 | intros. injection H0; intro; subst logn; clear H0. |
---|
| 1825 | assert (0 <= z < Z_of_nat wordsize). |
---|
| 1826 | apply H. auto with coqlib. |
---|
| 1827 | rewrite unsigned_repr. auto. generalize wordsize_max_unsigned; omega. |
---|
| 1828 | intros; discriminate. |
---|
| 1829 | Qed. |
---|
| 1830 | |
---|
| 1831 | Theorem is_power2_range: |
---|
| 1832 | forall n logn, |
---|
| 1833 | is_power2 n = Some logn -> ltu logn iwordsize = true. |
---|
| 1834 | Proof. |
---|
| 1835 | intros. unfold ltu. rewrite unsigned_repr_wordsize. |
---|
| 1836 | generalize (is_power2_rng _ _ H). |
---|
| 1837 | case (zlt (unsigned logn) (Z_of_nat wordsize)); intros. |
---|
| 1838 | auto. omegaContradiction. |
---|
| 1839 | Qed. |
---|
| 1840 | |
---|
| 1841 | Lemma is_power2_correct: |
---|
| 1842 | forall n logn, |
---|
| 1843 | is_power2 n = Some logn -> |
---|
| 1844 | unsigned n = two_p (unsigned logn). |
---|
| 1845 | Proof. |
---|
| 1846 | intros n logn. unfold is_power2. |
---|
| 1847 | generalize (Z_one_bits_powerserie (unsigned n) (unsigned_range n)). |
---|
| 1848 | generalize (Z_one_bits_range (unsigned n)). |
---|
| 1849 | destruct (Z_one_bits wordsize (unsigned n) 0). |
---|
| 1850 | intros; discriminate. |
---|
| 1851 | destruct l. |
---|
| 1852 | intros. simpl in H0. injection H1; intros; subst logn; clear H1. |
---|
| 1853 | rewrite unsigned_repr. replace (two_p z) with (two_p z + 0). |
---|
| 1854 | auto. omega. elim (H z); intros. |
---|
| 1855 | generalize wordsize_max_unsigned; omega. |
---|
| 1856 | auto with coqlib. |
---|
| 1857 | intros; discriminate. |
---|
| 1858 | Qed. |
---|
| 1859 | |
---|
| 1860 | Remark two_p_range: |
---|
| 1861 | forall n, |
---|
| 1862 | 0 <= n < Z_of_nat wordsize -> |
---|
| 1863 | 0 <= two_p n <= max_unsigned. |
---|
| 1864 | Proof. |
---|
| 1865 | intros. split. |
---|
| 1866 | assert (two_p n > 0). apply two_p_gt_ZERO. omega. omega. |
---|
| 1867 | generalize (two_p_monotone_strict _ _ H). rewrite <- two_power_nat_two_p. |
---|
| 1868 | unfold max_unsigned, modulus. omega. |
---|
| 1869 | Qed. |
---|
| 1870 | |
---|
| 1871 | Remark Z_one_bits_zero: |
---|
| 1872 | forall n i, Z_one_bits n 0 i = nil. |
---|
| 1873 | Proof. |
---|
| 1874 | induction n; intros; simpl; auto. |
---|
| 1875 | Qed. |
---|
| 1876 | |
---|
| 1877 | Remark Z_one_bits_two_p: |
---|
| 1878 | forall n x i, |
---|
| 1879 | 0 <= x < Z_of_nat n -> |
---|
| 1880 | Z_one_bits n (two_p x) i = (i + x) :: nil. |
---|
| 1881 | Proof. |
---|
| 1882 | induction n; intros; simpl. simpl in H. omegaContradiction. |
---|
| 1883 | rewrite inj_S in H. |
---|
| 1884 | assert (x = 0 \/ 0 < x) by omega. destruct H0. |
---|
| 1885 | subst x; simpl. decEq. omega. apply Z_one_bits_zero. |
---|
| 1886 | replace (two_p x) with (Z_shift_add false (two_p (x-1))). |
---|
| 1887 | rewrite Z_bin_decomp_shift_add. |
---|
| 1888 | replace (i + x) with ((i + 1) + (x - 1)) by omega. |
---|
| 1889 | apply IHn. omega. |
---|
| 1890 | unfold Z_shift_add. rewrite <- two_p_S. decEq; omega. omega. |
---|
| 1891 | Qed. |
---|
| 1892 | |
---|
| 1893 | Lemma is_power2_two_p: |
---|
| 1894 | forall n, 0 <= n < Z_of_nat wordsize -> |
---|
| 1895 | is_power2 (repr (two_p n)) = Some (repr n). |
---|
| 1896 | Proof. |
---|
| 1897 | intros. unfold is_power2. rewrite unsigned_repr. |
---|
| 1898 | rewrite Z_one_bits_two_p. auto. auto. |
---|
| 1899 | apply two_p_range. auto. |
---|
| 1900 | Qed. |
---|
| 1901 | |
---|
| 1902 | Theorem mul_pow2: |
---|
| 1903 | forall x n logn, |
---|
| 1904 | is_power2 n = Some logn -> |
---|
| 1905 | mul x n = shl x logn. |
---|
| 1906 | Proof. |
---|
| 1907 | intros. generalize (is_power2_correct n logn H); intro. |
---|
| 1908 | rewrite shl_mul_two_p. rewrite <- H0. rewrite repr_unsigned. |
---|
| 1909 | auto. |
---|
| 1910 | Qed. |
---|
| 1911 | |
---|
| 1912 | Lemma Z_of_bits_shift_rev: |
---|
| 1913 | forall n f, |
---|
| 1914 | (forall i, i >= Z_of_nat n -> f i = false) -> |
---|
| 1915 | Z_of_bits n f = Z_shift_add (f 0) (Z_of_bits n (fun i => f(i + 1))). |
---|
| 1916 | Proof. |
---|
| 1917 | induction n; intros. |
---|
| 1918 | simpl. rewrite H. reflexivity. unfold Z_of_nat. omega. |
---|
| 1919 | simpl. rewrite (IHn (fun i => f (i + 1))). |
---|
| 1920 | reflexivity. |
---|
| 1921 | intros. apply H. rewrite inj_S. omega. |
---|
| 1922 | Qed. |
---|
| 1923 | |
---|
| 1924 | Lemma Z_of_bits_shifts_rev: |
---|
| 1925 | forall m f, |
---|
| 1926 | 0 <= m -> |
---|
| 1927 | (forall i, i >= Z_of_nat wordsize -> f i = false) -> |
---|
| 1928 | exists k, |
---|
| 1929 | Z_of_bits wordsize f = k + two_p m * Z_of_bits wordsize (fun i => f(i + m)) |
---|
| 1930 | /\ 0 <= k < two_p m. |
---|
| 1931 | Proof. |
---|
| 1932 | intros. pattern m. apply natlike_ind. |
---|
| 1933 | exists 0. change (two_p 0) with 1. split. |
---|
| 1934 | transitivity (Z_of_bits wordsize (fun i => f (i + 0))). |
---|
| 1935 | apply Z_of_bits_exten. intros. decEq. omega. |
---|
| 1936 | omega. omega. |
---|
| 1937 | intros x POSx [k [EQ1 RANGE1]]. |
---|
| 1938 | set (f' := fun i => f (i + x)) in *. |
---|
| 1939 | assert (forall i, i >= Z_of_nat wordsize -> f' i = false). |
---|
| 1940 | intros. unfold f'. apply H0. omega. |
---|
| 1941 | generalize (Z_of_bits_shift_rev wordsize f' H1). intro. |
---|
| 1942 | rewrite EQ1. rewrite H2. |
---|
| 1943 | set (z := Z_of_bits wordsize (fun i => f (i + Zsucc x))). |
---|
| 1944 | replace (Z_of_bits wordsize (fun i => f' (i + 1))) with z. |
---|
| 1945 | rewrite two_p_S. |
---|
| 1946 | case (f' 0); unfold Z_shift_add. |
---|
| 1947 | exists (k + two_p x). split. ring. omega. |
---|
| 1948 | exists k. split. ring. omega. |
---|
| 1949 | auto. |
---|
| 1950 | unfold z. apply Z_of_bits_exten; intros. unfold f'. |
---|
| 1951 | decEq. omega. |
---|
| 1952 | auto. |
---|
| 1953 | Qed. |
---|
| 1954 | |
---|
| 1955 | Lemma shru_div_two_p: |
---|
| 1956 | forall x y, |
---|
| 1957 | shru x y = repr (unsigned x / two_p (unsigned y)). |
---|
| 1958 | Proof. |
---|
| 1959 | intros. unfold shru. |
---|
| 1960 | set (x' := unsigned x). set (y' := unsigned y). |
---|
| 1961 | elim (Z_of_bits_shifts_rev y' (bits_of_Z wordsize x')). |
---|
| 1962 | intros k [EQ RANGE]. |
---|
| 1963 | replace (Z_of_bits wordsize (bits_of_Z wordsize x')) with x' in EQ. |
---|
| 1964 | rewrite Zplus_comm in EQ. rewrite Zmult_comm in EQ. |
---|
| 1965 | generalize (Zdiv_unique _ _ _ _ EQ RANGE). intros. |
---|
| 1966 | rewrite H. auto. |
---|
| 1967 | apply eqm_small_eq. apply eqm_sym. apply Z_of_bits_of_Z. |
---|
| 1968 | unfold x'. apply unsigned_range. |
---|
| 1969 | auto with ints. |
---|
| 1970 | generalize (unsigned_range y). unfold y'. omega. |
---|
| 1971 | intros. apply bits_of_Z_above. auto. |
---|
| 1972 | Qed. |
---|
| 1973 | |
---|
| 1974 | Theorem shru_zero: |
---|
| 1975 | forall x, shru x zero = x. |
---|
| 1976 | Proof. |
---|
| 1977 | intros. rewrite shru_div_two_p. change (two_p (unsigned zero)) with 1. |
---|
| 1978 | transitivity (repr (unsigned x)). decEq. apply Zdiv_unique with 0. |
---|
| 1979 | omega. omega. auto with ints. |
---|
| 1980 | Qed. |
---|
| 1981 | |
---|
| 1982 | Theorem shr_zero: |
---|
| 1983 | forall x, shr x zero = x. |
---|
| 1984 | Proof. |
---|
| 1985 | intros. unfold shr. change (two_p (unsigned zero)) with 1. |
---|
| 1986 | replace (signed x / 1) with (signed x). |
---|
| 1987 | apply repr_signed. |
---|
| 1988 | symmetry. apply Zdiv_unique with 0. omega. omega. |
---|
| 1989 | Qed. |
---|
| 1990 | |
---|
| 1991 | Theorem divu_pow2: |
---|
| 1992 | forall x n logn, |
---|
| 1993 | is_power2 n = Some logn -> |
---|
| 1994 | divu x n = shru x logn. |
---|
| 1995 | Proof. |
---|
| 1996 | intros. generalize (is_power2_correct n logn H). intro. |
---|
| 1997 | symmetry. unfold divu. rewrite H0. apply shru_div_two_p. |
---|
| 1998 | Qed. |
---|
| 1999 | |
---|
| 2000 | Lemma modu_divu_Euclid: |
---|
| 2001 | forall x y, y <> zero -> x = add (mul (divu x y) y) (modu x y). |
---|
| 2002 | Proof. |
---|
| 2003 | intros. unfold add, mul, divu, modu. |
---|
| 2004 | transitivity (repr (unsigned x)). auto with ints. |
---|
| 2005 | apply eqm_samerepr. |
---|
| 2006 | set (x' := unsigned x). set (y' := unsigned y). |
---|
| 2007 | apply eqm_trans with ((x' / y') * y' + x' mod y'). |
---|
| 2008 | apply eqm_refl2. rewrite Zmult_comm. apply Z_div_mod_eq. |
---|
| 2009 | generalize (unsigned_range y); intro. |
---|
| 2010 | assert (unsigned y <> 0). red; intro. |
---|
| 2011 | elim H. rewrite <- (repr_unsigned y). unfold zero. congruence. |
---|
| 2012 | unfold y'. omega. |
---|
| 2013 | auto with ints. |
---|
| 2014 | Qed. |
---|
| 2015 | |
---|
| 2016 | Theorem modu_divu: |
---|
| 2017 | forall x y, y <> zero -> modu x y = sub x (mul (divu x y) y). |
---|
| 2018 | Proof. |
---|
| 2019 | intros. |
---|
| 2020 | assert (forall a b c, a = add b c -> c = sub a b). |
---|
| 2021 | intros. subst a. rewrite sub_add_l. rewrite sub_idem. |
---|
| 2022 | rewrite add_commut. rewrite add_zero. auto. |
---|
| 2023 | apply H0. apply modu_divu_Euclid. auto. |
---|
| 2024 | Qed. |
---|
| 2025 | |
---|
| 2026 | Theorem mods_divs: |
---|
| 2027 | forall x y, mods x y = sub x (mul (divs x y) y). |
---|
| 2028 | Proof. |
---|
| 2029 | intros; unfold mods, sub, mul, divs. |
---|
| 2030 | apply eqm_samerepr. |
---|
| 2031 | unfold Zmod_round. |
---|
| 2032 | apply eqm_sub. apply eqm_signed_unsigned. |
---|
| 2033 | apply eqm_unsigned_repr_r. |
---|
| 2034 | apply eqm_mult. auto with ints. apply eqm_signed_unsigned. |
---|
| 2035 | Qed. |
---|
| 2036 | |
---|
| 2037 | Theorem divs_pow2: |
---|
| 2038 | forall x n logn, |
---|
| 2039 | is_power2 n = Some logn -> |
---|
| 2040 | divs x n = shrx x logn. |
---|
| 2041 | Proof. |
---|
| 2042 | intros. generalize (is_power2_correct _ _ H); intro. |
---|
| 2043 | unfold shrx. rewrite shl_mul_two_p. |
---|
| 2044 | rewrite mul_commut. rewrite mul_one. |
---|
| 2045 | rewrite <- H0. rewrite repr_unsigned. auto. |
---|
| 2046 | Qed. |
---|
| 2047 | |
---|
| 2048 | Theorem shrx_carry: |
---|
| 2049 | forall x y, |
---|
| 2050 | add (shr x y) (shr_carry x y) = shrx x y. |
---|
| 2051 | Proof. |
---|
| 2052 | intros. unfold shr_carry. |
---|
| 2053 | rewrite sub_add_opp. rewrite add_permut. |
---|
| 2054 | rewrite add_neg_zero. apply add_zero. |
---|
| 2055 | Qed. |
---|
| 2056 | |
---|
| 2057 | Lemma Zdiv_round_Zdiv: |
---|
| 2058 | forall x y, |
---|
| 2059 | y > 0 -> |
---|
| 2060 | Zdiv_round x y = if zlt x 0 then (x + y - 1) / y else x / y. |
---|
| 2061 | Proof. |
---|
| 2062 | intros. unfold Zdiv_round. |
---|
| 2063 | destruct (zlt x 0). |
---|
| 2064 | rewrite zlt_false; try omega. |
---|
| 2065 | generalize (Z_div_mod_eq (-x) y H). |
---|
| 2066 | generalize (Z_mod_lt (-x) y H). |
---|
| 2067 | set (q := (-x) / y). set (r := (-x) mod y). intros. |
---|
| 2068 | symmetry. |
---|
| 2069 | apply Zdiv_unique with (y - r - 1). |
---|
| 2070 | replace x with (- (y * q) - r) by omega. |
---|
| 2071 | replace (-(y * q)) with ((-q) * y) by ring. |
---|
| 2072 | omega. |
---|
| 2073 | omega. |
---|
| 2074 | apply zlt_false. omega. |
---|
| 2075 | Qed. |
---|
| 2076 | |
---|
| 2077 | Theorem shrx_shr: |
---|
| 2078 | forall x y, |
---|
| 2079 | ltu y (repr (Z_of_nat wordsize - 1)) = true -> |
---|
| 2080 | shrx x y = |
---|
| 2081 | shr (if lt x zero then add x (sub (shl one y) one) else x) y. |
---|
| 2082 | Proof. |
---|
| 2083 | intros. unfold shrx, divs, shr. decEq. |
---|
| 2084 | exploit ltu_inv; eauto. rewrite unsigned_repr. |
---|
| 2085 | set (uy := unsigned y). |
---|
| 2086 | intro RANGE. |
---|
| 2087 | assert (shl one y = repr (two_p uy)). |
---|
| 2088 | transitivity (mul one (repr (two_p uy))). |
---|
| 2089 | symmetry. apply mul_pow2. replace y with (repr uy). |
---|
| 2090 | apply is_power2_two_p. omega. unfold uy. apply repr_unsigned. |
---|
| 2091 | rewrite mul_commut. apply mul_one. |
---|
| 2092 | assert (two_p uy > 0). apply two_p_gt_ZERO. omega. |
---|
| 2093 | assert (two_p uy < half_modulus). |
---|
| 2094 | rewrite half_modulus_power. |
---|
| 2095 | apply two_p_monotone_strict. auto. |
---|
| 2096 | assert (two_p uy < modulus). |
---|
| 2097 | rewrite modulus_power. apply two_p_monotone_strict. omega. |
---|
| 2098 | assert (unsigned (shl one y) = two_p uy). |
---|
| 2099 | rewrite H0. apply unsigned_repr. unfold max_unsigned. omega. |
---|
| 2100 | assert (signed (shl one y) = two_p uy). |
---|
| 2101 | rewrite H0. apply signed_repr. |
---|
| 2102 | unfold max_signed. generalize min_signed_neg. omega. |
---|
| 2103 | rewrite H5. |
---|
| 2104 | rewrite Zdiv_round_Zdiv; auto. |
---|
| 2105 | unfold lt. rewrite signed_zero. |
---|
| 2106 | destruct (zlt (signed x) 0); auto. |
---|
| 2107 | rewrite add_signed. |
---|
| 2108 | assert (signed (sub (shl one y) one) = two_p uy - 1). |
---|
| 2109 | unfold sub. rewrite H4. rewrite unsigned_one. |
---|
| 2110 | apply signed_repr. |
---|
| 2111 | generalize min_signed_neg. unfold max_signed. omega. |
---|
| 2112 | rewrite H6. rewrite signed_repr. decEq. omega. |
---|
| 2113 | generalize (signed_range x). intros. |
---|
| 2114 | assert (two_p uy - 1 <= max_signed). unfold max_signed. omega. |
---|
| 2115 | omega. |
---|
| 2116 | generalize wordsize_pos wordsize_max_unsigned; omega. |
---|
| 2117 | Qed. |
---|
| 2118 | |
---|
| 2119 | Lemma add_and: |
---|
| 2120 | forall x y z, |
---|
| 2121 | and y z = zero -> |
---|
| 2122 | add (and x y) (and x z) = and x (or y z). |
---|
| 2123 | Proof. |
---|
| 2124 | intros. unfold add, and, bitwise_binop. |
---|
| 2125 | decEq. |
---|
| 2126 | repeat rewrite unsigned_repr; auto with ints. |
---|
| 2127 | apply Z_of_bits_excl; intros. |
---|
| 2128 | assert (forall a b c, a && b && (a && c) = a && (b && c)). |
---|
| 2129 | destruct a; destruct b; destruct c; reflexivity. |
---|
| 2130 | rewrite H1. |
---|
| 2131 | replace (bits_of_Z wordsize (unsigned y) i && |
---|
| 2132 | bits_of_Z wordsize (unsigned z) i) |
---|
| 2133 | with (bits_of_Z wordsize (unsigned (and y z)) i). |
---|
| 2134 | rewrite H. change (unsigned zero) with 0. |
---|
| 2135 | rewrite bits_of_Z_zero. apply andb_b_false. |
---|
| 2136 | unfold and, bitwise_binop. |
---|
| 2137 | rewrite unsigned_repr; auto with ints. rewrite bits_of_Z_of_bits. |
---|
| 2138 | reflexivity. auto. |
---|
| 2139 | rewrite <- demorgan1. |
---|
| 2140 | unfold or, bitwise_binop. |
---|
| 2141 | rewrite unsigned_repr; auto with ints. rewrite bits_of_Z_of_bits; auto. |
---|
| 2142 | Qed. |
---|
| 2143 | |
---|
| 2144 | Lemma Z_of_bits_zero: |
---|
| 2145 | forall n f, |
---|
| 2146 | (forall i, i >= 0 -> f i = false) -> |
---|
| 2147 | Z_of_bits n f = 0. |
---|
| 2148 | Proof. |
---|
| 2149 | induction n; intros; simpl. |
---|
| 2150 | auto. |
---|
| 2151 | rewrite H. rewrite IHn. auto. intros. apply H. omega. omega. |
---|
| 2152 | Qed. |
---|
| 2153 | |
---|
| 2154 | Lemma Z_of_bits_trunc_1: |
---|
| 2155 | forall n f k, |
---|
| 2156 | (forall i, i >= k -> f i = false) -> |
---|
| 2157 | k >= 0 -> |
---|
| 2158 | 0 <= Z_of_bits n f < two_p k. |
---|
| 2159 | Proof. |
---|
| 2160 | induction n; intros. |
---|
| 2161 | simpl. assert (two_p k > 0). apply two_p_gt_ZERO; omega. omega. |
---|
| 2162 | destruct (zeq k 0). subst k. |
---|
| 2163 | change (two_p 0) with 1. rewrite Z_of_bits_zero. omega. auto. |
---|
| 2164 | simpl. replace (two_p k) with (2 * two_p (k - 1)). |
---|
| 2165 | assert (0 <= Z_of_bits n (fun i => f(i+1)) < two_p (k - 1)). |
---|
| 2166 | apply IHn. intros. apply H. omega. omega. |
---|
| 2167 | unfold Z_shift_add. destruct (f 0); omega. |
---|
| 2168 | rewrite <- two_p_S. decEq. omega. omega. |
---|
| 2169 | Qed. |
---|
| 2170 | |
---|
| 2171 | Lemma Z_of_bits_trunc_2: |
---|
| 2172 | forall n f1 f2 k, |
---|
| 2173 | (forall i, i < k -> f2 i = f1 i) -> |
---|
| 2174 | k >= 0 -> |
---|
| 2175 | exists q, Z_of_bits n f1 = q * two_p k + Z_of_bits n f2. |
---|
| 2176 | Proof. |
---|
| 2177 | induction n; intros. |
---|
| 2178 | simpl. exists 0; omega. |
---|
| 2179 | destruct (zeq k 0). subst k. |
---|
| 2180 | exists (Z_of_bits (S n) f1 - Z_of_bits (S n) f2). |
---|
| 2181 | change (two_p 0) with 1. omega. |
---|
| 2182 | destruct (IHn (fun i => f1 (i + 1)) (fun i => f2 (i + 1)) (k - 1)) as [q EQ]. |
---|
| 2183 | intros. apply H. omega. omega. |
---|
| 2184 | exists q. simpl. rewrite H. unfold Z_shift_add. |
---|
| 2185 | replace (two_p k) with (2 * two_p (k - 1)). rewrite EQ. |
---|
| 2186 | destruct (f1 0). ring. ring. |
---|
| 2187 | rewrite <- two_p_S. decEq. omega. omega. omega. |
---|
| 2188 | Qed. |
---|
| 2189 | |
---|
| 2190 | Lemma Z_of_bits_trunc_3: |
---|
| 2191 | forall f n k, |
---|
| 2192 | k >= 0 -> |
---|
| 2193 | Zmod (Z_of_bits n f) (two_p k) = Z_of_bits n (fun i => if zlt i k then f i else false). |
---|
| 2194 | Proof. |
---|
| 2195 | intros. |
---|
| 2196 | set (g := fun i : Z => if zlt i k then f i else false). |
---|
| 2197 | destruct (Z_of_bits_trunc_2 n f g k). |
---|
| 2198 | intros. unfold g. apply zlt_true. auto. |
---|
| 2199 | auto. |
---|
| 2200 | apply Zmod_unique with x. auto. |
---|
| 2201 | apply Z_of_bits_trunc_1. intros. unfold g. apply zlt_false. auto. auto. |
---|
| 2202 | Qed. |
---|
| 2203 | |
---|
| 2204 | Theorem modu_and: |
---|
| 2205 | forall x n logn, |
---|
| 2206 | is_power2 n = Some logn -> |
---|
| 2207 | modu x n = and x (sub n one). |
---|
| 2208 | Proof. |
---|
| 2209 | intros. generalize (is_power2_correct _ _ H); intro. |
---|
| 2210 | generalize (is_power2_rng _ _ H); intro. |
---|
| 2211 | unfold modu, and, bitwise_binop. |
---|
| 2212 | decEq. |
---|
| 2213 | set (ux := unsigned x). |
---|
| 2214 | replace ux with (Z_of_bits wordsize (bits_of_Z wordsize ux)). |
---|
| 2215 | rewrite H0. rewrite Z_of_bits_trunc_3. apply Z_of_bits_exten. intros. |
---|
| 2216 | rewrite bits_of_Z_of_bits; auto. |
---|
| 2217 | replace (unsigned (sub n one)) with (two_p (unsigned logn) - 1). |
---|
| 2218 | rewrite bits_of_Z_two_p. unfold proj_sumbool. |
---|
| 2219 | destruct (zlt z (unsigned logn)). rewrite andb_true_r; auto. rewrite andb_false_r; auto. |
---|
| 2220 | omega. auto. |
---|
| 2221 | rewrite <- H0. unfold sub. symmetry. rewrite unsigned_one. apply unsigned_repr. |
---|
| 2222 | rewrite H0. |
---|
| 2223 | assert (two_p (unsigned logn) > 0). apply two_p_gt_ZERO. omega. |
---|
| 2224 | generalize (two_p_range _ H1). omega. |
---|
| 2225 | omega. |
---|
| 2226 | apply eqm_small_eq. apply Z_of_bits_of_Z. apply Z_of_bits_range. |
---|
| 2227 | unfold ux. apply unsigned_range. |
---|
| 2228 | Qed. |
---|
| 2229 | |
---|
| 2230 | (** ** Properties of integer zero extension and sign extension. *) |
---|
| 2231 | |
---|
| 2232 | Section EXTENSIONS. |
---|
| 2233 | |
---|
| 2234 | Variable n: Z. |
---|
| 2235 | Hypothesis RANGE: 0 < n < Z_of_nat wordsize. |
---|
| 2236 | |
---|
| 2237 | Remark two_p_n_pos: |
---|
| 2238 | two_p n > 0. |
---|
| 2239 | Proof. apply two_p_gt_ZERO. omega. Qed. |
---|
| 2240 | |
---|
| 2241 | Remark two_p_n_range: |
---|
| 2242 | 0 <= two_p n <= max_unsigned. |
---|
| 2243 | Proof. apply two_p_range. omega. Qed. |
---|
| 2244 | |
---|
| 2245 | Remark two_p_n_range': |
---|
| 2246 | two_p n <= max_signed + 1. |
---|
| 2247 | Proof. |
---|
| 2248 | unfold max_signed. rewrite half_modulus_power. |
---|
| 2249 | assert (two_p n <= two_p (Z_of_nat wordsize - 1)). |
---|
| 2250 | apply two_p_monotone. omega. |
---|
| 2251 | omega. |
---|
| 2252 | Qed. |
---|
| 2253 | |
---|
| 2254 | Remark unsigned_repr_two_p: |
---|
| 2255 | unsigned (repr (two_p n)) = two_p n. |
---|
| 2256 | Proof. |
---|
| 2257 | apply unsigned_repr. apply two_p_n_range. |
---|
| 2258 | Qed. |
---|
| 2259 | |
---|
| 2260 | Theorem zero_ext_and: |
---|
| 2261 | forall x, zero_ext n x = and x (repr (two_p n - 1)). |
---|
| 2262 | Proof. |
---|
| 2263 | intros; unfold zero_ext. |
---|
| 2264 | assert (is_power2 (repr (two_p n)) = Some (repr n)). |
---|
| 2265 | apply is_power2_two_p. omega. |
---|
| 2266 | generalize (modu_and x _ _ H). |
---|
| 2267 | unfold modu. rewrite unsigned_repr_two_p. intro. rewrite H0. |
---|
| 2268 | decEq. unfold sub. decEq. rewrite unsigned_repr_two_p. |
---|
| 2269 | rewrite unsigned_one. reflexivity. |
---|
| 2270 | Qed. |
---|
| 2271 | |
---|
| 2272 | Theorem zero_ext_idem: |
---|
| 2273 | forall x, zero_ext n (zero_ext n x) = zero_ext n x. |
---|
| 2274 | Proof. |
---|
| 2275 | intros. repeat rewrite zero_ext_and. |
---|
| 2276 | rewrite and_assoc. rewrite and_idem. auto. |
---|
| 2277 | Qed. |
---|
| 2278 | |
---|
| 2279 | Lemma eqm_eqmod_two_p: |
---|
| 2280 | forall a b, eqm a b -> eqmod (two_p n) a b. |
---|
| 2281 | Proof. |
---|
| 2282 | intros a b [k EQ]. |
---|
| 2283 | exists (k * two_p (Z_of_nat wordsize - n)). |
---|
| 2284 | rewrite EQ. decEq. rewrite <- Zmult_assoc. decEq. |
---|
| 2285 | rewrite <- two_p_is_exp. unfold modulus. rewrite two_power_nat_two_p. |
---|
| 2286 | decEq. omega. omega. omega. |
---|
| 2287 | Qed. |
---|
| 2288 | |
---|
| 2289 | Lemma sign_ext_charact: |
---|
| 2290 | forall x y, |
---|
| 2291 | -(two_p (n-1)) <= signed y < two_p (n-1) -> |
---|
| 2292 | eqmod (two_p n) (unsigned x) (signed y) -> |
---|
| 2293 | sign_ext n x = y. |
---|
| 2294 | Proof. |
---|
| 2295 | intros. unfold sign_ext. set (x' := unsigned x) in *. |
---|
| 2296 | destruct H0 as [k EQ]. |
---|
| 2297 | assert (two_p n = 2 * two_p (n - 1)). rewrite <- two_p_S. decEq. omega. omega. |
---|
| 2298 | assert (signed y >= 0 \/ signed y < 0) by omega. destruct H1. |
---|
| 2299 | assert (x' mod two_p n = signed y). |
---|
| 2300 | apply Zmod_unique with k; auto. omega. |
---|
| 2301 | rewrite zlt_true. rewrite H2. apply repr_signed. omega. |
---|
| 2302 | assert (x' mod two_p n = signed y + two_p n). |
---|
| 2303 | apply Zmod_unique with (k-1). rewrite EQ. ring. omega. |
---|
| 2304 | rewrite zlt_false. replace (x' mod two_p n - two_p n) with (signed y) by omega. apply repr_signed. |
---|
| 2305 | omega. |
---|
| 2306 | Qed. |
---|
| 2307 | |
---|
| 2308 | Lemma zero_ext_eqmod_two_p: |
---|
| 2309 | forall x y, |
---|
| 2310 | eqmod (two_p n) (unsigned x) (unsigned y) -> zero_ext n x = zero_ext n y. |
---|
| 2311 | Proof. |
---|
| 2312 | intros. unfold zero_ext. decEq. apply eqmod_mod_eq. apply two_p_n_pos. auto. |
---|
| 2313 | Qed. |
---|
| 2314 | |
---|
| 2315 | Lemma sign_ext_eqmod_two_p: |
---|
| 2316 | forall x y, |
---|
| 2317 | eqmod (two_p n) (unsigned x) (unsigned y) -> sign_ext n x = sign_ext n y. |
---|
| 2318 | Proof. |
---|
| 2319 | intros. unfold sign_ext. |
---|
| 2320 | assert (unsigned x mod two_p n = unsigned y mod two_p n). |
---|
| 2321 | apply eqmod_mod_eq. apply two_p_n_pos. auto. |
---|
| 2322 | rewrite H0. auto. |
---|
| 2323 | Qed. |
---|
| 2324 | |
---|
| 2325 | Lemma eqmod_two_p_zero_ext: |
---|
| 2326 | forall x, eqmod (two_p n) (unsigned x) (unsigned (zero_ext n x)). |
---|
| 2327 | Proof. |
---|
| 2328 | intros. unfold zero_ext. |
---|
| 2329 | apply eqmod_trans with (unsigned x mod two_p n). |
---|
| 2330 | apply eqmod_mod. apply two_p_n_pos. |
---|
| 2331 | apply eqm_eqmod_two_p. apply eqm_unsigned_repr. |
---|
| 2332 | Qed. |
---|
| 2333 | |
---|
| 2334 | Lemma eqmod_two_p_sign_ext: |
---|
| 2335 | forall x, eqmod (two_p n) (unsigned x) (unsigned (sign_ext n x)). |
---|
| 2336 | Proof. |
---|
| 2337 | intros. unfold sign_ext. destruct (zlt (unsigned x mod two_p n) (two_p (n-1))). |
---|
| 2338 | apply eqmod_trans with (unsigned x mod two_p n). |
---|
| 2339 | apply eqmod_mod. apply two_p_n_pos. |
---|
| 2340 | apply eqm_eqmod_two_p. apply eqm_unsigned_repr. |
---|
| 2341 | apply eqmod_trans with (unsigned x mod two_p n). |
---|
| 2342 | apply eqmod_mod. apply two_p_n_pos. |
---|
| 2343 | apply eqmod_trans with (unsigned x mod two_p n - 0). |
---|
| 2344 | apply eqmod_refl2. omega. |
---|
| 2345 | apply eqmod_trans with (unsigned x mod two_p n - two_p n). |
---|
| 2346 | apply eqmod_sub. apply eqmod_refl. exists (-1). ring. |
---|
| 2347 | apply eqm_eqmod_two_p. apply eqm_unsigned_repr. |
---|
| 2348 | Qed. |
---|
| 2349 | |
---|
| 2350 | Theorem sign_ext_idem: |
---|
| 2351 | forall x, sign_ext n (sign_ext n x) = sign_ext n x. |
---|
| 2352 | Proof. |
---|
| 2353 | intros. apply sign_ext_eqmod_two_p. |
---|
| 2354 | apply eqmod_sym. apply eqmod_two_p_sign_ext. |
---|
| 2355 | Qed. |
---|
| 2356 | *) |
---|
[487] | 2357 | axiom sign_ext_zero_ext: |
---|
[3] | 2358 | ∀n:Z.∀RANGE: 0 < n ∧ n < wordsize.∀x. sign_ext n (zero_ext n x) = sign_ext n x. |
---|
| 2359 | (* |
---|
| 2360 | Theorem sign_ext_zero_ext: |
---|
| 2361 | forall x, sign_ext n (zero_ext n x) = sign_ext n x. |
---|
| 2362 | Proof. |
---|
| 2363 | intros. apply sign_ext_eqmod_two_p. |
---|
| 2364 | apply eqmod_sym. apply eqmod_two_p_zero_ext. |
---|
| 2365 | Qed. |
---|
| 2366 | |
---|
| 2367 | Theorem zero_ext_sign_ext: |
---|
| 2368 | forall x, zero_ext n (sign_ext n x) = zero_ext n x. |
---|
| 2369 | Proof. |
---|
| 2370 | intros. apply zero_ext_eqmod_two_p. |
---|
| 2371 | apply eqmod_sym. apply eqmod_two_p_sign_ext. |
---|
| 2372 | Qed. |
---|
| 2373 | *) |
---|
[487] | 2374 | axiom sign_ext_equal_if_zero_equal: |
---|
[3] | 2375 | ∀n:Z.∀RANGE: 0 < n ∧ n < wordsize.∀x,y. |
---|
| 2376 | zero_ext n x = zero_ext n y -> |
---|
| 2377 | sign_ext n x = sign_ext n y. |
---|
| 2378 | (* |
---|
| 2379 | Theorem sign_ext_equal_if_zero_equal: |
---|
| 2380 | forall x y, |
---|
| 2381 | zero_ext n x = zero_ext n y -> |
---|
| 2382 | sign_ext n x = sign_ext n y. |
---|
| 2383 | Proof. |
---|
| 2384 | intros. rewrite <- (sign_ext_zero_ext x). |
---|
| 2385 | rewrite <- (sign_ext_zero_ext y). congruence. |
---|
| 2386 | Qed. |
---|
| 2387 | |
---|
| 2388 | Lemma eqmod_mult_div: |
---|
| 2389 | forall n1 n2 x y, |
---|
| 2390 | 0 <= n1 -> 0 <= n2 -> |
---|
| 2391 | eqmod (two_p (n1+n2)) (two_p n1 * x) y -> |
---|
| 2392 | eqmod (two_p n2) x (y / two_p n1). |
---|
| 2393 | Proof. |
---|
| 2394 | intros. rewrite two_p_is_exp in H1; auto. |
---|
| 2395 | destruct H1 as [k EQ]. exists k. |
---|
| 2396 | change x with (0 / two_p n1 + x). rewrite <- Z_div_plus. |
---|
| 2397 | replace (0 + x * two_p n1) with (two_p n1 * x) by ring. |
---|
| 2398 | rewrite EQ. |
---|
| 2399 | replace (k * (two_p n1 * two_p n2) + y) with (y + (k * two_p n2) * two_p n1) by ring. |
---|
| 2400 | rewrite Z_div_plus. ring. |
---|
| 2401 | apply two_p_gt_ZERO; auto. |
---|
| 2402 | apply two_p_gt_ZERO; auto. |
---|
| 2403 | Qed. |
---|
| 2404 | |
---|
| 2405 | Theorem sign_ext_shr_shl: |
---|
| 2406 | forall x, |
---|
| 2407 | let y := repr (Z_of_nat wordsize - n) in |
---|
| 2408 | sign_ext n x = shr (shl x y) y. |
---|
| 2409 | Proof. |
---|
| 2410 | intros. |
---|
| 2411 | assert (unsigned y = Z_of_nat wordsize - n). |
---|
| 2412 | unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega. |
---|
| 2413 | apply sign_ext_charact. |
---|
| 2414 | (* inequalities *) |
---|
| 2415 | unfold shr. rewrite H. |
---|
| 2416 | set (z := signed (shl x y)). |
---|
| 2417 | rewrite signed_repr. |
---|
| 2418 | apply Zdiv_interval_1. |
---|
| 2419 | assert (two_p (n - 1) > 0). apply two_p_gt_ZERO. omega. omega. |
---|
| 2420 | apply two_p_gt_ZERO. omega. |
---|
| 2421 | apply two_p_gt_ZERO. omega. |
---|
| 2422 | replace ((- two_p (n-1)) * two_p (Z_of_nat wordsize - n)) |
---|
| 2423 | with (- (two_p (n-1) * two_p (Z_of_nat wordsize - n))) by ring. |
---|
| 2424 | rewrite <- two_p_is_exp. |
---|
| 2425 | replace (n - 1 + (Z_of_nat wordsize - n)) with (Z_of_nat wordsize - 1) by omega. |
---|
| 2426 | rewrite <- half_modulus_power. |
---|
| 2427 | generalize (signed_range (shl x y)). unfold z, min_signed, max_signed. omega. |
---|
| 2428 | omega. omega. |
---|
| 2429 | apply Zdiv_interval_2. unfold z. apply signed_range. |
---|
| 2430 | generalize min_signed_neg; omega. generalize max_signed_pos; omega. |
---|
| 2431 | apply two_p_gt_ZERO; omega. |
---|
| 2432 | (* eqmod *) |
---|
| 2433 | unfold shr. rewrite H. |
---|
| 2434 | apply eqmod_trans with (signed (shl x y) / two_p (Z_of_nat wordsize - n)). |
---|
| 2435 | apply eqmod_mult_div. omega. omega. |
---|
| 2436 | replace (Z_of_nat wordsize - n + n) with (Z_of_nat wordsize) by omega. |
---|
| 2437 | rewrite <- two_power_nat_two_p. |
---|
| 2438 | change (eqm (two_p (Z_of_nat wordsize - n) * unsigned x) (signed (shl x y))). |
---|
| 2439 | rewrite shl_mul_two_p. unfold mul. rewrite H. |
---|
| 2440 | apply eqm_sym. eapply eqm_trans. apply eqm_signed_unsigned. |
---|
| 2441 | apply eqm_unsigned_repr_l. rewrite (Zmult_comm (unsigned x)). |
---|
| 2442 | apply eqm_mult. apply eqm_sym. apply eqm_unsigned_repr. apply eqm_refl. |
---|
| 2443 | apply eqm_eqmod_two_p. apply eqm_sym. eapply eqm_trans. |
---|
| 2444 | apply eqm_signed_unsigned. apply eqm_sym. apply eqm_unsigned_repr. |
---|
| 2445 | Qed. |
---|
| 2446 | |
---|
| 2447 | Theorem zero_ext_shru_shl: |
---|
| 2448 | forall x, |
---|
| 2449 | let y := repr (Z_of_nat wordsize - n) in |
---|
| 2450 | zero_ext n x = shru (shl x y) y. |
---|
| 2451 | Proof. |
---|
| 2452 | intros. |
---|
| 2453 | assert (unsigned y = Z_of_nat wordsize - n). |
---|
| 2454 | unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega. |
---|
| 2455 | rewrite zero_ext_and. symmetry. |
---|
| 2456 | replace n with (Z_of_nat wordsize - unsigned y). |
---|
| 2457 | apply shru_shl_and. unfold ltu. apply zlt_true. |
---|
| 2458 | rewrite H. rewrite unsigned_repr_wordsize. omega. omega. |
---|
| 2459 | Qed. |
---|
| 2460 | |
---|
| 2461 | End EXTENSIONS. |
---|
| 2462 | |
---|
| 2463 | (** ** Properties of [one_bits] (decomposition in sum of powers of two) *) |
---|
| 2464 | |
---|
| 2465 | Opaque Z_one_bits. (* Otherwise, next Qed blows up! *) |
---|
| 2466 | |
---|
| 2467 | Theorem one_bits_range: |
---|
| 2468 | forall x i, In i (one_bits x) -> ltu i iwordsize = true. |
---|
| 2469 | Proof. |
---|
| 2470 | intros. unfold one_bits in H. |
---|
| 2471 | elim (list_in_map_inv _ _ _ H). intros i0 [EQ IN]. |
---|
| 2472 | subst i. unfold ltu. unfold iwordsize. apply zlt_true. |
---|
| 2473 | generalize (Z_one_bits_range _ _ IN). intros. |
---|
| 2474 | assert (0 <= Z_of_nat wordsize <= max_unsigned). |
---|
| 2475 | generalize wordsize_pos wordsize_max_unsigned; omega. |
---|
| 2476 | repeat rewrite unsigned_repr; omega. |
---|
| 2477 | Qed. |
---|
| 2478 | |
---|
| 2479 | Fixpoint int_of_one_bits (l: list int) : int := |
---|
| 2480 | match l with |
---|
| 2481 | | nil => zero |
---|
| 2482 | | a :: b => add (shl one a) (int_of_one_bits b) |
---|
| 2483 | end. |
---|
| 2484 | |
---|
| 2485 | Theorem one_bits_decomp: |
---|
| 2486 | forall x, x = int_of_one_bits (one_bits x). |
---|
| 2487 | Proof. |
---|
| 2488 | intros. |
---|
| 2489 | transitivity (repr (powerserie (Z_one_bits wordsize (unsigned x) 0))). |
---|
| 2490 | transitivity (repr (unsigned x)). |
---|
| 2491 | auto with ints. decEq. apply Z_one_bits_powerserie. |
---|
| 2492 | auto with ints. |
---|
| 2493 | unfold one_bits. |
---|
| 2494 | generalize (Z_one_bits_range (unsigned x)). |
---|
| 2495 | generalize (Z_one_bits wordsize (unsigned x) 0). |
---|
| 2496 | induction l. |
---|
| 2497 | intros; reflexivity. |
---|
| 2498 | intros; simpl. rewrite <- IHl. unfold add. apply eqm_samerepr. |
---|
| 2499 | apply eqm_add. rewrite shl_mul_two_p. rewrite mul_commut. |
---|
| 2500 | rewrite mul_one. apply eqm_unsigned_repr_r. |
---|
| 2501 | rewrite unsigned_repr. auto with ints. |
---|
| 2502 | generalize (H a (in_eq _ _)). generalize wordsize_max_unsigned. omega. |
---|
| 2503 | auto with ints. |
---|
| 2504 | intros; apply H; auto with coqlib. |
---|
| 2505 | Qed. |
---|
| 2506 | |
---|
| 2507 | (** ** Properties of comparisons *) |
---|
| 2508 | |
---|
| 2509 | Theorem negate_cmp: |
---|
| 2510 | forall c x y, cmp (negate_comparison c) x y = negb (cmp c x y). |
---|
| 2511 | Proof. |
---|
| 2512 | intros. destruct c; simpl; try rewrite negb_elim; auto. |
---|
| 2513 | Qed. |
---|
| 2514 | |
---|
| 2515 | Theorem negate_cmpu: |
---|
| 2516 | forall c x y, cmpu (negate_comparison c) x y = negb (cmpu c x y). |
---|
| 2517 | Proof. |
---|
| 2518 | intros. destruct c; simpl; try rewrite negb_elim; auto. |
---|
| 2519 | Qed. |
---|
| 2520 | |
---|
| 2521 | Theorem swap_cmp: |
---|
| 2522 | forall c x y, cmp (swap_comparison c) x y = cmp c y x. |
---|
| 2523 | Proof. |
---|
| 2524 | intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym. |
---|
| 2525 | Qed. |
---|
| 2526 | |
---|
| 2527 | Theorem swap_cmpu: |
---|
| 2528 | forall c x y, cmpu (swap_comparison c) x y = cmpu c y x. |
---|
| 2529 | Proof. |
---|
| 2530 | intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym. |
---|
| 2531 | Qed. |
---|
| 2532 | |
---|
| 2533 | Lemma translate_eq: |
---|
| 2534 | forall x y d, |
---|
| 2535 | eq (add x d) (add y d) = eq x y. |
---|
| 2536 | Proof. |
---|
| 2537 | intros. unfold eq. case (zeq (unsigned x) (unsigned y)); intro. |
---|
| 2538 | unfold add. rewrite e. apply zeq_true. |
---|
| 2539 | apply zeq_false. unfold add. red; intro. apply n. |
---|
| 2540 | apply eqm_small_eq; auto with ints. |
---|
| 2541 | replace (unsigned x) with ((unsigned x + unsigned d) - unsigned d). |
---|
| 2542 | replace (unsigned y) with ((unsigned y + unsigned d) - unsigned d). |
---|
| 2543 | apply eqm_sub. apply eqm_trans with (unsigned (repr (unsigned x + unsigned d))). |
---|
| 2544 | eauto with ints. apply eqm_trans with (unsigned (repr (unsigned y + unsigned d))). |
---|
| 2545 | eauto with ints. eauto with ints. eauto with ints. |
---|
| 2546 | omega. omega. |
---|
| 2547 | Qed. |
---|
| 2548 | |
---|
| 2549 | Lemma translate_lt: |
---|
| 2550 | forall x y d, |
---|
| 2551 | min_signed <= signed x + signed d <= max_signed -> |
---|
| 2552 | min_signed <= signed y + signed d <= max_signed -> |
---|
| 2553 | lt (add x d) (add y d) = lt x y. |
---|
| 2554 | Proof. |
---|
| 2555 | intros. repeat rewrite add_signed. unfold lt. |
---|
| 2556 | repeat rewrite signed_repr; auto. case (zlt (signed x) (signed y)); intro. |
---|
| 2557 | apply zlt_true. omega. |
---|
| 2558 | apply zlt_false. omega. |
---|
| 2559 | Qed. |
---|
| 2560 | |
---|
| 2561 | Theorem translate_cmp: |
---|
| 2562 | forall c x y d, |
---|
| 2563 | min_signed <= signed x + signed d <= max_signed -> |
---|
| 2564 | min_signed <= signed y + signed d <= max_signed -> |
---|
| 2565 | cmp c (add x d) (add y d) = cmp c x y. |
---|
| 2566 | Proof. |
---|
| 2567 | intros. unfold cmp. |
---|
| 2568 | rewrite translate_eq. repeat rewrite translate_lt; auto. |
---|
| 2569 | Qed. |
---|
| 2570 | |
---|
| 2571 | Theorem notbool_isfalse_istrue: |
---|
| 2572 | forall x, is_false x -> is_true (notbool x). |
---|
| 2573 | Proof. |
---|
| 2574 | unfold is_false, is_true, notbool; intros; subst x. |
---|
| 2575 | simpl. apply one_not_zero. |
---|
| 2576 | Qed. |
---|
| 2577 | |
---|
| 2578 | Theorem notbool_istrue_isfalse: |
---|
| 2579 | forall x, is_true x -> is_false (notbool x). |
---|
| 2580 | Proof. |
---|
| 2581 | unfold is_false, is_true, notbool; intros. |
---|
| 2582 | generalize (eq_spec x zero). case (eq x zero); intro. |
---|
| 2583 | contradiction. auto. |
---|
| 2584 | Qed. |
---|
| 2585 | |
---|
| 2586 | Theorem shru_lt_zero: |
---|
| 2587 | forall x, |
---|
| 2588 | shru x (repr (Z_of_nat wordsize - 1)) = if lt x zero then one else zero. |
---|
| 2589 | Proof. |
---|
| 2590 | intros. rewrite shru_div_two_p. |
---|
| 2591 | replace (two_p (unsigned (repr (Z_of_nat wordsize - 1)))) |
---|
| 2592 | with half_modulus. |
---|
| 2593 | generalize (unsigned_range x); intro. |
---|
| 2594 | unfold lt. rewrite signed_zero. unfold signed. |
---|
| 2595 | destruct (zlt (unsigned x) half_modulus). |
---|
| 2596 | rewrite zlt_false. |
---|
| 2597 | replace (unsigned x / half_modulus) with 0. reflexivity. |
---|
| 2598 | symmetry. apply Zdiv_unique with (unsigned x). ring. omega. omega. |
---|
| 2599 | rewrite zlt_true. |
---|
| 2600 | replace (unsigned x / half_modulus) with 1. reflexivity. |
---|
| 2601 | symmetry. apply Zdiv_unique with (unsigned x - half_modulus). ring. |
---|
| 2602 | rewrite half_modulus_modulus in H. omega. omega. |
---|
| 2603 | rewrite unsigned_repr. apply half_modulus_power. |
---|
| 2604 | generalize wordsize_pos wordsize_max_unsigned; omega. |
---|
| 2605 | Qed. |
---|
| 2606 | |
---|
| 2607 | Theorem ltu_range_test: |
---|
| 2608 | forall x y, |
---|
| 2609 | ltu x y = true -> unsigned y <= max_signed -> |
---|
| 2610 | 0 <= signed x < unsigned y. |
---|
| 2611 | Proof. |
---|
| 2612 | intros. |
---|
| 2613 | unfold ltu in H. destruct (zlt (unsigned x) (unsigned y)); try discriminate. |
---|
| 2614 | rewrite signed_eq_unsigned. |
---|
| 2615 | generalize (unsigned_range x). omega. omega. |
---|
| 2616 | Qed. |
---|
| 2617 | |
---|
| 2618 | End Make. |
---|
| 2619 | |
---|
| 2620 | (** * Specialization to 32-bit integers. *) |
---|
| 2621 | |
---|
| 2622 | Module IntWordsize. |
---|
| 2623 | Definition wordsize := 32%nat. |
---|
| 2624 | Remark wordsize_not_zero: wordsize <> 0%nat. |
---|
| 2625 | Proof. unfold wordsize; congruence. Qed. |
---|
| 2626 | End IntWordsize. |
---|
| 2627 | |
---|
| 2628 | Module Int := Make(IntWordsize). |
---|
| 2629 | |
---|
| 2630 | Notation int := Int.int. |
---|
| 2631 | |
---|
| 2632 | Remark int_wordsize_divides_modulus: |
---|
| 2633 | Zdivide (Z_of_nat Int.wordsize) Int.modulus. |
---|
| 2634 | Proof. |
---|
| 2635 | exists (two_p (32-5)); reflexivity. |
---|
| 2636 | Qed. |
---|
| 2637 | *) |
---|
| 2638 | |
---|
| 2639 | |
---|
[747] | 2640 | *) |
---|